Analytical Chemistry Theoretical and Metrological Fundamentals K

Analytical Chemistry Theoretical and Metrological Fundamentals K. Danzer Analytical Chemistry

Theoretical and Metrological Fundamentals

With 138 Figures and 31 Tables K. Danzer Am Friedensberg 4 07745 Jena Germany e-mail: [email protected]

Library of Congress Control Number 2006932265

ISBN-10 3-540-35988-5 Springer Berlin Heidelberg New York ISBN-13 978-3-540-35988-3 Springer Berlin Heidelberg New York DOI 10.1007/b103950

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Analytical chemistry can look back on a long history. In the course of its development, analytical chemistry has contributed essentially to the progress of diverse ˇelds of science and technology. Signiˇcant progress in natural philosophy, chemistry and other disciplines has been founded on results and discoveries in analytical chemistry. Well known examples include Archimedes' principle, iatrochemistry, emergence and overcoming of phlo- giston theory, basic chemical laws on stoichiometry and mass proportions as well as the discoveries of elements and of nuclear ˇssion. As chemistry split into inorganic, organic and physical chemistry in the middle of the nineteenth century, the question occasionally arose as to whether analytical chemistry was likewise an autonomous ˇeld of chemistry. At the beginning of the twentieth century, analytical chemistry was established by Wilhelm Ostwald and other protagonists on the basis of chemical and physicochemical laws and principles. In addition, analytical procedures and instruments spread into other chemical ˇelds. However, in the 1950s and 1960s a new generation of analysts opened their mind to ideas coming from other scientiˇc ˇelds dealing with measurements from other points of view such as information science, metrology, technometrics, etc. As a result, efforts made towards the acceptance of analytical chemistry as an autonomous ˇeld of chemistry increased. Analysts in USA and Europe founded organisations and forums that focussed ideas and developments in progressive directions. In this way, not only were national and international societies formed but also regular conferences like Pittcon and Euroanalysis. In this connection the so-called Lindau circle should be mentioned where analysts from Germany, Austria and Switzerland made public relevant fundamentals of analytical chemistry derived from statistics, information theory, system theory, signal theory, game theory, decision theory, metrology, etc. In parallel with this, such modern principles have been successfully applied in selected case studies, mainly by scientists from USA, UK, and Canada. From this development, real theoretical and metrological fundamentals of analytical chemistry have crystallized. However, the acceptance of analytical chemistry as an automomous branch of chemistry has not been realised. Gradually, however, progress in analytical chemistry culminated in the 1990s in the proclamation of a new scientiˇc discipline, analytical science. Protagonists of these activities included Michael Widmer and Klaus Doerffel. VI Preface Today, analytical chemistry is still a discipline within chemistry. Al- though characterized as an auxiliary science, analytical chemistry continues to develop and grow just as before. It is no detraction being characterized as an auxiliary discipline as mathematics shows. However, it is likely that efforts to make analytical chemistry a more independent science will be repeated in the future from time to time. Analytical chemistry possesses today a sound basis of chemical, physical, methodical, metrological, and theoretical fundamentals. The ˇrst of these are usually taken as the basis of classical textbooks on analytical chemistry. The others are found in diverse publications in the ˇeld of analytical chemistry and chemometrics. It is essential to state that chemometrics is not the theoretical basis of analytical chemistry but it contributes signiˇcantly to it. Frequently, analytical chemistry is considered to be a measuring science in chemistry. Therefore, its object is the generation, evaluation, interpretation, and validation of measuring signals as well as the characterization of their uncertainty. With this aim, the analyst needs knowledge of the general analytical process, statistics, optimization, calibration, chemometrical data analysis, and performance characteristics. In this book the attempt is made to summarize all the components that can be considered as building blocks of a theory of analytical chemistry. The \building" constructed in this way is a provisional one. It is incomplete and, therefore, extension, reconstruction and rebuilding have to be expected in the future. A large number of mathematical formulas will be found in the book. This may be regarded as a disadvantage particularly because some of them are not readily to apply in daily analytical practice. However, great scientists, explicitly Emanuel Kant, said that a scientiˇc branch contains only so much of science as it applies mathematics. Consequently, all the relationships which can be described mathematically should be so described. It is true despite Werner Heisenbergs statement: Although natural processes can be described by means of simple laws which can be precisely formulated, these laws, on the other hand, cannot directly be applied to actions in practice. Wherever possible, ofˇcial deˇnitions of IUPAC, ISO and other international organizations have been used, in particular in the \Glossary of Analytical Terms" compiled at the end of the book. However, uniformity could not be achieved in every case. In a few instances, special comments and proposals (characterized as such) have been added. Although progress in the ˇeld of harmonization of nomenclature and deˇnitions has been con- siderable, some things still remain to be done. Some of the contents of various sections have been published previously and are, of course quoted verbatim. In this connection the author is grateful to Klaus Doerffel, Karel Eckschlager, Gunter Ehrlich, Andrzej Parczewski, Karol Florian, Mikulas Matherny, Gunter Marx, Dieter Molch, Eberhard Than, Ludwig Kuchler and others for long-standing collaboration resulting in mutually complementary papers and books. A chapter on statistics was previously published in another book (Accreditation and Quality Assurance Preface VII in Analytical Chemistry edited by Helmut Gunzler at Springer). Gratefully I have adapted essential parts of the translation of Gaida Lapitajs here in this book. Stimulations and ideas have arisen from discussions with William Hor- witz, Duncan Thorburn Burns, Alan Townshend, Koos van Staden, and other members of diverse IUPAC Commissions and Task Groups as well as ofˇ- cers of the Analytical Chemistry Division. From a 20-year membership of various IUPAC bodies I have gained a variety of feedback and experience which has proved to be useful in the writing of this book. In particular, I owe Lloyd Currie and Mattias Otto a great debt of gratitude. They have permitted me to use essential parts of common publications on calibration quoted in Chap. 6. Many of the results stem from work carried out with colleagues, collaborators, postdocs and graduate students. All of them are deserving of my thanks but, due to the large numbers involved, this cannot be done here. Some representative colleagues include Jurgen Einax, Hart- mut Hobert, Werner Schron, Manfred Reichenbacher, Reiner Singer, Diet- rich Wienke, Christoph Fischbacher, Kai-Uwe Jagemann, Michael Wagner, Katrin Venth, Raimund Horn, Gabriela Thiel, Demetrio de la Calle Garca, and Balint Berente. The author gratefully acknowledges the cooperation with and the support of the editorial staff of Springer. Particularly I have to thank Peter Enders, Birgit Kollmar-Thoni, and { last but not least { John Kirby, the English copy editor, who has essentially improved the English. Many thanks for that. No book is free of errors and this one will be no exception. Therefore, the author would be grateful to readers who point out errors and mistakes and suggest any improvement.

Jena, July 2006 Klaus Danzer Contents

Symbols ...... XIII

Abbreviations and Acronyms ...... XXV

1 Object of Analytical Chemistry ...... 1 1.1 Deˇnition of Analytical Chemistry ...... 1 1.2 Repertoire of Analytical Chemistry ...... 5 References ...... 10

2 The Analytical Process ...... 13 2.1 Principles of Sampling ...... 15 2.2 Sample Preparation ...... 23 2.3 Principles of Analytical Measurement ...... 26 2.4 Analytical Evaluation ...... 31 References ...... 37

3 Signals in Analytical Chemistry ...... 43 3.1 Signals and Information ...... 43 3.2 Analytical Signals ...... 44 3.3 Types and Properties of Analytical Signals ...... 47 3.4 Dimensionality of Analytical Signals and Information ...... 53 3.5 Mathematical Model of Signal Generation...... 60 References ...... 63

4 Statistical Evaluation of Analytical Results ...... 65 4.1 Reliability of Analytical Observations and Measurements . . . 65 4.1.1 Systematic Deviations ...... 67 4.1.2 Random Variations ...... 69 4.2 Uncertainty Concept ...... 75 4.3 Statistical Tests ...... 78 4.3.1 Null Hypotheses ...... 79 4.3.2 Test for Measurement Series ...... 80 4.3.3 Comparison of Standard Deviations ...... 81 4.3.4 Comparison of Measured Values ...... 82 X Contents 4.4 Reliability of Qualitative Analytical Tests ...... 85 4.5 Statistical Quality Control ...... 90 4.5.1 Quality Criteria for Analytical Results ...... 90 4.5.2 Attribute Testing ...... 92 4.5.3 Sequential Analysis ...... 93 4.5.4 Statistical Quality Control ...... 95 References ...... 98

5 Studying Inuences and Optimizing Analytical Procedures ...... 101 5.1 Testing the Signiˇcance of Inuencing Factors ...... 101 5.1.1 Analysis of Variance (ANOVA) ...... 101 5.1.2 Experimental Design ...... 108 5.2 Optimization of Analytical Procedures ...... 112 5.3 Global Optimization by Natural Design ...... 116 References ...... 120

6 Calibration in Analytical Chemistry ...... 123 6.1 General Fundamentals of Calibration ...... 124 6.1.1 Fundamental and Experimental Calibration ...... 124 6.1.2 The General Three-Dimensional Calibration Model ...... 126 6.1.3 Regression and Calibration ...... 127 6.2 Single Component Calibration ...... 130 6.2.1 Linear Calibration Model ...... 130 6.2.2 Errors in Linear Calibration ...... 134 6.2.3 Weighted Linear Least Squares Estimation (WLS) ...... 137 6.2.4 Linear Least Squares Fitting in Case of Errors in Both Variables ...... 138 6.2.5 Statistical Tests and Validation of Calibration ...... 140 6.2.6 Alternative Calibration Procedures ...... 144 6.2.7 Nonlinear Calibration ...... 151 6.3 Multisignal Calibration ...... 152 6.4 Multicomponent Calibration ...... 155 6.4.1 Classical Multivariate Calibration ...... 157 6.4.2 Inverse Calibration ...... 159 6.4.3 Validation of Multivariate Calibration ...... 162 6.5 Calibration by Artiˇcial Neural Networks ...... 165 References ...... 172

7 Analytical Performance Characteristics ...... 177 7.1 Reliability of Analytical Measurements ...... 178 7.1.1 Precision ...... 178 7.1.2 Precision of Trace Analyses ...... 182 7.1.3 Accuracy and Trueness ...... 183 Contents XI 7.1.4 Remark on the Quantiˇcation of Precision, Accuracy and Trueness ...... 183 7.2 Sensitivity ...... 185 7.3 Selectivity and Speciˇcity ...... 189 7.4 Robustness and Ruggedness ...... 195 7.5 Limit Values ...... 201 7.6 Resolving Power ...... 209 References ...... 212

8 Presentation, Interpretation and Validation of Analytical Results 217 8.1 Presentation of Analytical Results ...... 217 8.2 Factual Interpretation of Analytical Results ...... 219 8.2.1 Presentation of Results Near the Limit of Detection ...... 219 8.2.2 Missing Data ...... 221 8.2.3 Analytical Results in Relation to Fixed Values ...... 224 8.2.4 Interlaboratory Studies ...... 227 8.3 Chemometrical Interpretation of Analytical Data ...... 228 8.3.1 Principles of Data Analysis ...... 229 8.3.2 Cluster Analysis: Recognition of Inherent Data Structures . . 231 8.3.3 Classiˇcation: Modelling of Data Structures ...... 235 8.3.4 Factor Analysis: Causes of Data Structures ...... 239 8.3.5 Exploratory Data Analysis and Display Methods: Visualization of Data Structures ...... 243 8.3.6 Methods of Artiˇcial Intelligence ...... 246 8.4 Analytical Images ...... 250 References ...... 257

9 Assessment of Analytical Information ...... 265 9.1 Quantiˇcation of Information ...... 265 9.2 Information Content of Quantitative Analysis ...... 268 9.3 Multicomponent Analysis ...... 273 9.4 Process and Image Analysis ...... 276 References ...... 280

Glossary of Analytical Terms ...... 283 References ...... 305

Index ...... 309 Symbols

A Matrix of sensitivity factors in (6.66) multicomponent analysis (e.g., absorbance factors) a Regression coefˇcient (calibration coefˇcient): intercept (general and regression in case of errors in both variables) ax Regression coefˇcient a for the esti- (6.9) mation of y from x axw Regression coefˇcient a in case of (6.37) weighted calibration ay Regression coefˇcient a for the esti- (6.10) mation of x from y acc(x), acc(x) Accuracy of an analytical result x (7.10) and of a mean x, respectively (7.11) antilg x =10x , antilogarithm of x b Regression coefˇcient (calibration coefˇcient): slope (general and in case of regression having errors in both variables) bx Regression coefˇcient b for the esti- (6.8) mation of y from x bxw Regression coefˇcient b in case of (6.36) weighted calibration by Regression coefˇcient b for the esti- (6.10) mation of x from y bias(x), bias(y ) Bias (systematic deviation) of an in- (4.1) dividual test value x (or y ) from the (conventional) true value xtrue (or ytrue) XIV Symbols bias(x), bias(y ) Bias of an average test value x (or y ) from the (conventional) true value xtrue (or ytrue)

cnf(x), cnf(y ) Total (two-sided) conˇdence inter- (4.17) vall of a mean cond(A) Condition number of the matrix A (6.80)

cov(pi;pj ) Covariance of the quantities pi and pj CB Conˇdence band (of a calibration (6.26a) line) CR Concordance rate (proportion of (4.53a) correct test results in screening)

dcrit Critical diameter (in microprobe analysis)

dij Distance (in multivariate data) (8.12) df Discriminant function (8.16) dv Discriminant variable

E Information efˇciency

Ein Input energy

Eout Output energy E Residual matrix (in multivariate data analysis)

e, ex , ey Error term (general, in x, and in y , respectively), in concrete terms, ey may be the deviation of an individual value from the mean or the residual of a mathematical model, see next line (ex analogous)

ey = yi y , deviation of an individual measured value from the mean (also dy )

ey = yi y^, deviation of an individual measured value from the estimate of the corresponding mathematical model

F^ Test statistic (estimate) of the F test (4.37) Symbols XV

F1˛;1;2 Quantile of the F distribution at the level of signiˇcance 1˛ and for the degrees of freedom 1 and 2 0 F^max Test statistic (estimate) of Hartley's (4.37 ) test FNR False negative rate (of screening (4.51) tests) FPR False positive rate (of screening (4.50) tests) fdet(:::) Deterministic function of . . . femp(:::) Empirical function of . . .

G^ Test statistic (estimate) of Grubb's (4.36) outlier test

G^max Test statistic (estimate) of Coch- (4.38) ran's test

G˛;n Signiˇcance limits of Grubb's outlier test for a risk of error ˛ and n individual measurements gaccept(n) Acceptance function in attribute testing greject(n) Rejection function in attribute testing gaccept(n;s;˛) Acceptance function in variable testing gaccept(n;s;ˇ) Rejection function in variable testing

H 0 a posteriori information entropy H a priori information entropy

H0 Null hypothesis

HA Alternative hypothesis hom(A) Quantity characterizing the homo- (2.9) geneity of an analyte A in a sample

I Information content (9.1)

IAj Speciˇc strength of an inuence factor j on the signal of analyte A

I(p;p0) Divergence measure of information (9.9)

J Information ow (9.40) XVI Symbols L Loading matrix (of principal compo- (6.44) nent analysis and factor analysis) (8.19) LCL Lower control limit (in quality control) LWL Lower warning limit (in quality control)

lx , ly , lz Dimensions (lengths) in x-, y - and z-direction, respectively lb a Binary logarithm of a

M Information amount M(n) Information amount of multicompo- (9.21) nent analysis (of n components)

madfyig Median absolute deviation (around the median), i.e., the median of the differences between an individual measured value and the median of the series: madfyig = medfjyi medfyigjg

medfyig Median of a set of measured values (4.22)

N Number of repeated measurements for the estimation of x by means of a calibration function

Npp Noise amplitude, peak-to-peak noise N (;2) Normal distribution with mean and variance 2 N (t) Time average of noise NPR Negative prediction rate (in screen- (4.55) ing tests) n Number of individual measurements (also in establishing a calibration model), number of objects (in a data set) n(t) Noise component of a signal function in the time domain (i) neti(t), net Net function (propagation function) (6.117) in a neural network (layer i)

(i) Oi(t), O Output of a layer i of a neural net- (6.119) work Symbols XVII P Two-sided level of signiˇcance of statistical tests (P =1 ˛) P One-sided level of signiˇcance of statistical tests (P =1 ˛) P (A) Probability of an event A (ratio of the number of times the event occurs to the total number of trials) P (A) Probability that the analyte A is present in the test sample P (A) Probability of the opposite of the event A (ratio of the number of times the event does not occur to the total number, P (A) + P (A)=1) P (A) Probability that the analyte A is present in the test sample P (AjB) Conditional probability: probability of an event B on the condition that another event A occurs P (T +jA) Probability that the analyte A is present in the test sample if a test result T is positive P Score matrix (of principal compo- (6.44) nent analysis) PB Prediction band (of a calibration (6.26b) line) PPR Positive prediction rate (in screening (4.54) tests) PV Prevalency of screening tests: proba- (4.52) bility that the analyte A is present in a given number of samples p Number of primary samples (in sampling); Number of calibration points pi, pj , pij Inuence parameters (4.25) prec(x) Precision of an analytical result x (7.8) prd(x), prd(a) Prediction interval of a quantity x (and a, respectively) q Number of subsamples (in sampling) (2.5) q^R Test statistic (estimate) of the David (4.33) test XVIII Symbols

Q Categorial variable characterizing chemical entities (species) being investigated, e.g., elements, isotopes, ions, compounds Q^ Test statistic (estimate) of Dixon's (4.35) outlier test Q(z) Evaluation relationship (of quantities characterizing qualitative properties) in form of a function, atlas, or table Q = f 1(z) Evaluation function of quantities characterizing qualitative properties, viz type of species, Q, in dependence of signal position, z

R Range (R = ymax ymin) R Resolution power R Correlation matrix (6.4)

RC Risk for customers (in quality assur- (4.56b) ance)

RM Redundancy (9.31a)

RM Risk for manufacturers (in quality (4.56a) assurance)

Rt Temporal resolution power (7.57)

Rz Analytical resolution power (7.53) RMSP Root mean standard error of predic- (6.128) tion RR Recovery rate Rfqg Range of a quantity q r Number of measurements at a sub- (2.4) sample

rM Relative redundancy (9.31b)

rxy Correlation coefˇcient between the (6.3) (random) variables x and y

rob(A=B:::; f1 :::) Robustness of a procedure to deter- (7.31) mine an analyte A with regard to disturbing components (B;:::) and factors (f1;:::) Symbols XIX rug(A=B:::; f1 :::; u1 :::) Ruggedness of a procedure to deter- (7.33) mine an analyte A with regard to disturbing components (B;:::), factors (f1;:::), and unknowns (u1;:::)

S Sensitivity, derivative of the measured quantity (response) y with respect to the analytical quantity x

Stotal Total multicomponent sensitivity (7.18) S Covariance matrix (6.5) Sensitivity matrix (7.17)

SAA Sensitivity of the response of the an- (7.12) alyte A with respect to the amount xA of the analyte A (also SA)

SAB Cross sensitivity (partial sensitivity) of the response of the analyte A with respect to the amount xB of the component B

SAi Cross sensitivity (partial sensitivity) (3.11) of the response of the analyte A with respect to the amount xi of species i (i = B;C;:::N) 0 Sxx, Syy Sum of squared deviations (of x and (6.3 ) y , respectively) 0 Sxy Sum of crossed deviations (of x and (6.3 ) y )

S, Si Sum over a variable index (e.g., at constant i); the dot replaces the index over that the summation is carried out S(x) Sensitivity function (6.64) s Estimate of the standard deviation (4.12) (SD of the sample) s2 Estimate of the variance (variance of (4.10) the sample) sN Standard deviation of noise (7.5) spipj , s(pi;pj ) Covariance of the quantities pi (4.27) and pj sy:x Residual standard deviation (of the (6.19) calibration); estimate of the error of the calibration model XX Symbols sel(A;B;:::N) Selectivity of a multicomponent (7.24) analysis with regard to the analytes A;B;:::N snr(y ) Signal-to-noise ratio of an average (7.6) signal intensity y (related to Npp) spec(A=B;:::;N) Speciˇcity of the determination of an (7.26) analyte A with regard to the accompanying components B;:::N S=N Signal-to-noise ratio of an average (7.1) signal intensity y or a net intensity y net (related to sy ) (S=N)c Critical signal-to-noise ratio (7.51) SS Sum of squares ssr(y ) Sum of squares of residuals (of y ) (6.12)

tW Test statistic (estimate) of the gener- (4.43) alized t test (Welch test) TE Test efˇciency of screening tests (4.53b) TNR True negative rate (of screening (4.49) tests) TPR True positive rate (of screening tests) (4.48) t^ Test statistic (estimate) of Student's (4.41) t test

t1˛; Quantile of the t-distribution at the level of signiˇcance 1 ˛ and for degrees of freedom

UCL Upper control limit (in quality control) UWL Upper warning limit (in quality control) u(x) Combined uncertainty of an analyti- (4.31) cal value x

u(x(p1;p2;:::)) Uncertainty of an analytical value x, combined from the uncertainties of the parameters p1;p2;::: u(y ) Combined uncertainty of a measured value y

u(y (p1;p2;:::)) Uncertainty of a measured value y , (4.25) combined from the uncertainties of the parameters p1;p2;::: Symbols XXI

U (x) Extended combined uncertainty (limit) of an analytical value x U (y ) Extended combined uncertainty (4.29) (limit) of a measured value y unc(x) Uncertainty interval of a mean x (4.32) unc(y ) Uncertainty interval of a mean y (4.30)

wyi , wi Weight coefˇcient (in weighted cali- (6.34) bration) x Analytical value: analyte amount, e.g., content, concentration xLD Limit of detection (7.44) xLQ Limit of quantitation (7.48) xQ Analytical value (amount) of a species Q xtest Analytical value of a test sample xtrue (Conventional) true value of a (cer- tiˇed) reference sample x Arithmetic mean of x (of the sample) x^ Estimate of x x(Q) Sample composition (function) x = f (Q) Analytical function in the sample domain, sample composition function

Y (!) Signal function in the frequency domain, Fourier transform of y (t) y Measured value, response (e.g., intensity of a signal) yc Critical value (7.41) yzi Measured value (e.g. intensity) of a signal at position zi y Arithmetic mean of y (of the sample) y Total mean of several measured series y BL Blank y geom Geometric mean of y (of the sample) y^ Estimate of y XXII Symbols y Outlier-suspected value among the measured values y (t) Signal function in the time domain

ytrue(t) Component of the signal function in the time domain that is considered being true (inuenced by noise) y (z) Signal record (spectrum, chromatogram, etc.) y = f (x) Calibration function (in the stricter sense, i.e. of quantities characterizing quantitative properties, e.g., signal intensity vs analyte amount) y = f (z) Signal function (measurement function, analytical function in the signal domain)

z Signal position: measuring quantity that depends on a qualitative property of the measurand. Therefore, analytes may be identiˇed by characteristic signal positions. The z-scale may be directly or reciprocally proportional to an energy quantity or time zz-scores, standardized analytical val- (8.9) ues z(Q) Signal assignment function (table, atlas etc.) z = f (Q) Calibration function of quantities characterizing qualitative properties, viz signal position as a function of the type of species

˛ Risk of the error of the ˇrst kind (two-sided) ˛ Regression coefˇcient (intercept) (4.2)

˛i Coefˇcient characterizing inuences (5.3) ˛ Risk of the error of the ˇrst kind (one-sided)

ˇ Risk of the error of the second kind (two-sided) Symbols XXIII ˇ Regression coefˇcient (slope) (4.3)

ˇj Coefˇcient characterizing inuences (5.4) ˇ Risk of the error of the second kind (one-sided)

^2 Test statistic (estimate) of the 2 test 2 2 1˛; Quantile of the distribution at the signiˇcance level 1 ˛ and for degrees of freedom

xcnf Conˇdence limit of a mean x

xprd Prediction limit of a mean x

y cnf Conˇdence limit of a mean y 2 Test statistic (estimate) for von Neu- (4.34) mann's trend test

Ffx(t)g Fourier transform of the time function x(t) into the frequency function X(!) F 1fX(!)g Fourier backtransform of the frequency function X(!) into the time function x(t)

Exponent characterizing nonlinear (4.5) errors Eigenvalue (characteristic root) of a matrix

M Multiplet splitting according to relevant rules

x Mean of x of the population

y Mean of y of the population

Number of the statistical degrees of freedom Dispersion factor (4.20)

Standard deviation (of the population) 2 Variance (of the population) (4.8)

Kaiser's selectivity (7.21) XXIV Symbols

A Kaiser's speciˇcity with regard to (7.22) an analyte A

xx() Autocorrelation function (ACF) of a (2.11) function x(t) with time lag

(˛ˇ)ij Coefˇcient characterizing interac- (5.4) tions of inuences ˛ and ˇ

(˛ˇ)ijk Coefˇcient characterizing interac- (5.5) tions of inuences ˛, ˇ, and

fxig Set (series, sequence) of analytical results whose terms are x1;x2;:::;xi;:::

fyig Set of measured values (observations of a measurement series) whose terms are y1;y2;:::;yi;:::

a;^ b;:::^ Estimates of the quantities a;b;:::

In general, variables are expressed by italics, vectors by bold small letters, and matrices by bold capital letters. Abbreviations and Acronyms

This list of abbreviations contains both acronyms which are generally used in analytical chemistry and such applied in the book. In addition to terms from analytical methods, essential statistical and chemometrical terms as well as acronyms of institutions and organizations are included. Terms of very particular interest are explained on that spot. 2D Two-Dimensional (e.g., 2D-NMR) 3D Three-Dimensional AA Activation Analysis AAS Atomic Absorption Spectrometry ACF Autocorrelation Function ACV Analytical value at critical (measuring) value AEM Analytical Electron Microscopy AES Auger Electron Spectroscopy AES (Atomic Emission Spectrometry) ! OAES, OES AFM Atomic Force Microscopy AFS Atomic Fluorescence Spectrometry ANN Artiˇcial Neural Networks ANOVA Analysis Of Variance AOAC Association of Ofˇcial Analytical Chemists ARM Atomic Resolution Microscopy ARUPS Angle Resolved Ultraviolet Photoelectron Spectrometry ASV Anodic Stripping Voltammetry ATR Attenuated Total Reectance BASIC Programming language: Beginners All-purpose Symbolic BCA Beckman Glucose Analyzer BIPM Bureau International des Ponds et Mesures CA Chemical Analysis CARS Coherent Anti-Stokes Raman Spectrometry CCC Counter Current Chromatography CCD Charge-Coupled Device CE Capillary Electrophoresis XXVI Abbreviations and Acronyms C{E{R Calibration-Evaluation-Recovery (Function) CFA Continuous Flow Analysis CGC Capillary Gas Chromatography CMP Chemical Measurement Process CPAA Charged Particle Activation Analysis CRM Certiˇed Reference Material CV Critical value CV-AAS Cold Vapour Atomic Absorption Spectrometry DCM Dielectric Constant Measurement (Dielcometry) Dendral Expert system: Dendritic Algorithm DIN Deutsches Institut fur Normung DSC Differential Scanning Calorimetry DTA Differential Thermal Analysis DTG Differential Thermogravimetry EBV Errors in Both Variables (Model, Procedure) ECA Electrochemical Analysis ECD Electron Capture Detector ED Electron Diffraction EDX Energy-Dispersive X-Ray (Spectrometry) ED-XFA Energy-Dispersive X-Ray Fluorescence Analysis EDL Electrodeless Discharge Lamp EELS Electron Energy Loss Spectrometry EM Electron Microscopy EMP Electron Microprobe EN European Norm EPA Environmental Protection Agency (USA) EPH Electrophoresis EPMA Electron Probe Microanalysis ESAC Expert Systems in Analytical Chemistry ESCA Electron Spectroscopy for Chemical Analysis ESD Estimated Standard Deviation ESR Electron Spin Resonance (Spectroscopy) ETA Electrothermal Atomizer ETA-AAS Electrothermal Atomizing Atomic Absorption Spectrometry EXAFS Extended X-Ray Absorption Fine Structure (Spectrometry) FAB Fast Atom Bombardment FANES Furnace Atomization Non-Thermal Emission Spectrometry FD Field Desorption FEM Field Electron Microscopy Abbreviations and Acronyms XXVII FFT Fast Fourier Transform FIA Flow Injection Analysis FI-AP Field Ion Atom Probe FID Flame-Ionization Detector FIM Field Ion Microscopy fn false negative (decisions in screening tests) fp false positive (decisions in screening tests) FNAA Fast Neutron Activation Analysis Fortran Programming language: Formula Translation FT Fourier Transform FT-ICR-MS Fourier Transform Ion Cyclotron Resonance Mass Spectrom- etry FT-IR, FTIR Fourier Transform Infrared (Spectrometry) FT-MS Fourier Transform Mass Spectrometry FT-NMR Fourier Transform Nuclear Magnetic Resonance Spectrome- try FWHM Full Width at Half Maximum GA Genetic Algorithm GC Gas Chromatography GC-IR Gas Chromatography Infrared Spectrometry Coupling GC-MS Gas Chromatography Mass Spectrometry Coupling GDL Glow Discharge Lamp GDMS Glow Discharge Mass Spectrometry GDOS Glow Discharge Optical Spectrometry GF-AAS Graphite Furnace Atomic Absorption Spectrometry GLC Gas Liquid Chromatography GLS Gaussian least squares (regression) GLP Good Laboratory Practice GMP Good Manufacturing Practice HCL Hollow Cathode Lamp HEED High Energy Electron Diffraction HEIS High Energy Ion Scattering HG-AAS Hydride Generation Atomic Absorption Spectrometry HPLC High Performance Liquid Chromatography HPTLC High Performance Thin-Layer Chromatography HR High Resolution HRMS High Resolution Mass Spectrometry IC Ion Chromatography ICP Inductively Coupled Plasma (Spectrometry) XXVIII Abbreviations and Acronyms ICP-MS Inductively Coupled Plasma Mass Spectrometry ICP-OES Inductively Coupled Plasma Optical Emission Spectrometry ICR Ion Cyclotron Resonance ID Ion Diffraction IDMS Isotope Dilution Mass Spectrometry IEC Ion Exchange Chromatography IEC International Electrotechnical Commission IFCC International Federation of Clinical Chemistry IM Ion Microscopy IMS Ion Mobility Spectrometry INAA Instrumental Neutron Activation Analysis INDOR Internuclear Double Resonance INS Inelastic Neutron Scattering IR, IRS Infrared Spectroscopy IRM Infrared Microscopy ISE Ion-Selective Electrodes IRS Internal Reectance Spectroscopy ISO International Organisation for Standardization ISS Ion Surface Scattering, Ion Scattering Spectrometry IU Insulin unit IUPAC International Union of Pure and Applied Chemistry IUPAP International Union of Pure and Applied Physics LAMMS Laser (Ablation) Micro Mass Spectrometry LASER Light Ampliˇcation by Stimulated Emission of Radiation LC Liquid Chromatography LC-MS Liquid Chromatography Mass Spectrometry Coupling LD Limit of Detection LDMS Laser Desorption Mass Spectrometry LEED Low Energy Electron Diffraction LEIS Low Energy Ion Scattering LEMS Laser Excited Mass Spectrometry LIDAR Light Detection And Ranging (analogous to RADAR) LIMS Laboratory Information Management Systems LISP ListProcessing LLC Liquid Liquid Chromatography LM Light Microscopy LMA Laser Microspectral Analysis LM-OES Laser Micro Optical Emission Spectroscopy LOD Limit Of Detection Abbreviations and Acronyms XXIX LOS, LoS Level of signiˇcance LQ Limit of quantitation LRMA Laser Raman Micro Analysis MALDI Matrix-Assisted Laser Desorption and Ionization MALDI-TOF MALDI Time-Of-Flight (Mass Spectrometry) MEIS Medium Energy Ion Scattering MES Mossbauer Effect Spectroscopy MIP Microwave-Induced Plasma MLD Measured value at Limit of Detection MLQ Measured value at Limit of Quantitation MORD Magneto Optical Rotary Dispersion MOS Mossbauer Spectrometry MS Mass Spectrometry MS-MS Tandem Mass Spectrometry MSn n-fold Tandem Mass Spectrometry M-OES Micro(spark) Optical Emission Spectroscopy MWS Microwave Spectroscopy NAA Neutron Activation Analysis NBS National Bureau of Standards, USA (today: NIST) ND Neutron Diffraction NEXAFS Near-Edge X-Ray Absorption Fine Structure (Spectroscopy) NIR, NIRS Near Infrared (Spectrometry) NIST National Institute of Standards and Technology, USA NMR Nuclear Magnetic Resonance (Spectroscopy) NOE Nuclear Overhauser Effect NQR Nuclear Quadrupole Resonance OAES Optical Atomic Emission Spectroscopy OCS Out-of-control situations (in quality control) OES Optical Emission Spectroscopy OIML Organisation Internationale de Metrologie Legale OLS Ordinary least squares (regression) ORD Optical Rotary Dispersion PARC Pattern Recognition PAS Photo Acoustic Spectroscopy PASCAL Programming language called by Blaise Pascal (1623{1662) PC Paper Chromatography PCR Principal component regression PES Photoelectron Spectrometry PIXE Particle Induced X-Ray Emission XXX Abbreviations and Acronyms PLS Partial least squares (regression) PMT Photomultiplier Tube ppm part per million (106 corresponding to 104 per cent) ppb part per billion (109 corresponding to 107 per cent) ppt part per trillion (1012 corresponding to 1010 per cent) ppq part per quatrillion (1015 corresponding to 1013 per cent) PROLOG Programming language: Programming in Logic QCC Quality control charts QMS Quadropol Mass Spectrometer R&D Research and Development RADAR Radiowave Detection And Ranging RBS Rutherford Backscattering Spectrometry REELS Reection Electron Energy Loss Spectrometry REM Reection Electron Microscopy RHEED Reection High Energy Electron Diffraction RF Refractometry RIMS Resonance Ionization Mass Spectrometry RM Reference Material RMSP Root mean standard error of prediction RPC Reversed Phase Chromatography RPLC Reversed Phase Liquid Chromatography RRS Resonance Raman Scattering RS Raman Spectroscopy SAM Scanning Auger Microscopy SAM Standard Addition Method SEC Size Exclusion Chromatography SEM Scanning Electron Microscopy SERS Surface Enhanced Raman Scattering SFC Supercritical Fluid Chromatography SI Systeme International (d'Unites) SIMS Secondary Ion Mass Spectrometry SNMS Sputtered Neutrals Mass Spectrometry SNR Signal-to-Noise Ratio SOP Standard Operating Procedure SPE Solid Phase Extraction SPME Solid Phase Micro Extraction SRM Standard Reference Material SSMS Spark Source Mass Spectrometry STEM Scanning Transmission Electron Microscopy Abbreviations and Acronyms XXXI STM Scanning Tunneling Microscopy SV Standard value TA Thermal Analysis TCD Thermal Conductivity Detector TEELS Transmission Electron Energy Loss Spectrometry TEM Transmission Electron Microscopy TG Thermogravimetry TGA Thermogravimetric Analysis THEED Transmission High Energy Electron Diffraction TIC Total Ion Chromatogram TID Thermoionic Detector TLC Thin-Layer Chromatography TMA Thermomechanical Analysis tn true negative (decisions in screening tests) TOF-(MS) Time-Of-Flight (Mass Spectrometry) tp true positive (decisions in screening tests) TXRF Total Reection X-ray Fluorescence (Spectrometry) UPS Ultraviolet Photoelectron Spectrometry UV Ultraviolet (radiation) UV-VIS Ultraviolet-Visible (Spectrometry) VIS Visible (radiation) WD-XFA Wavelength-Dispersive X-Ray Fluorescence (Spectrometry) XAS X-Ray Absorption Spectrometry XD X-Ray Diffraction XFA X-Ray Fluorescence Analysis XFS X-Ray Fluorescence Spectrometry XPS X-Ray Photoelectron Spectrometry XRD X-Ray Diffraction XRF X-Ray Fluorescence (Spectrometry) ZAF Z (stands for atomic number) Absorption Fluorescence (Cor- rection) (XFA) ZAAS Zeeman Atomic Absorption Spectrometry 1 Object of Analytical Chemistry

Analytical chemistry is one of the oldest scientiˇc disciplines. Its history can be traced back to the ancient Egyptians, about four to ˇve thousand years ago (Szabadvary [1966], Malissa [1987], Yordanov [1987]). But notwithstanding its long history, analytical chemistry is always an essential factor in the development of modern scientiˇc and industrialised society. The development of chemistry itself has progressed signiˇcantly by analytical ˇndings over several centuries. Fundamental knowledge of general chemistry is based on analytical studies, the laws of simple and multiple proportions as well as the law of mass action. Most of the chemical elements have been discovered by the application of analytical chemistry, at ˇrst by means of chemical methods, but in the last 150 years mainly by physical methods. Especially spectacular were the spectroscopic discoveries of rubidium and caesium by Bunsen and Kirchhoff, indium by Reich and Richter, helium by Janssen, Lockyer, and Frankland, and rhenium by Noddack and Tacke. Also, nuclear ˇssion became evident as Hahn and Strassmann carefully analyzed the products of neutron-bombarded uranium. In recent times, analytical chemistry has stimulated not only chemistry but many ˇelds of science, technology and society. Conversely, analytical chemistry itself has always been heavily inuenced by ˇelds like nuclear engineering, materials science, environmental protection, biology, and medicine. Figure 1.1 shows by which challenges analytical chemistry has been stimulated to improved performances within the last half century. Wilhelm Ostwald [1894], who published the ˇrst comprehensive textbook on analytical chemistry, emphasized in it the service function of analytical chemistry. This fact has not changed until now. Interactions with all the ˇelds of application have always had a promoting inuence on analytical chemistry.

1.1 Deˇnition of Analytical Chemistry

In the second half of the twentieth century, analytical chemistry was deˇned as the chemical discipline that gains information on the chemical composition and structure of substances, particularly on the type of species, their amount, 2 1 Object of Analytical Chemistry

Fig. 1.1. Economic and social challenges and stimuli for the development of some important branches of analytical chemistry

possible temporal and spatial changes, and structural relationships between the constituents (see, e.g. Eckschlager and Danzer [1994]; Danzer et al. [1976]). Sometimes the development of methods and instruments is included or is central to such deˇnitions, e.g. that of WPAC [1993]: \Analytical chemistry is found to be a scientiˇc discipline that develops and applies methods, instruments, and strategies to obtain information on the composition and nature of matter in space and time" (Kellner et al. [1998]). However, now and then analytical chemists feel uneasy with such kinds of deˇnitions which do not reect completely the identity and independence of analytical chemistry. Chemists of other branches (inorganic, organic, and physical chemists) as well as physicists and bioscientists also obtain information on inanimate or living matter using and developing high-performance analytical instruments just as analytical chemists do. Consequently, there exists a wide range of diverse deˇnitions. One of the most appropriate is that of Reilley [1965]: \Analytical chemistry is what analytical chemists do" which is, however, not really helpful. The ˇrst deˇnition that is focused directly on the role of analytical signals was given by Pungor who characterizes analytical chemistry as \a science of signal production and interpretation" (Veress et al. [1987], Lewenstam and Zytkow [1987]). Zolotov [1984] characterized chemical, physicochemical and physical methods of analytical chemistry as follows: \All of them, however, have the same feature: it is the dependence of signal on analyte 1.1 Deˇnition of Analytical Chemistry 3 concentration. The important task of analytical chemistry is therefore the discovery and implantation of these dependencies into analytical procedures" (Lewenstam and Zytkow [1987]). To overcome the unsatisfactory situation in the understanding the meaning of analytical chemistry at the end of the last century, an international competition was organized in 1992 by noted European analytical chemists and the Fresenius Journal of Analytical Chemistry to characterize analytical chemistry as an autonomous ˇeld of science by a topical and proper deˇni- tion. The title of this competition was \Analytical Chemistry { today's deˇ- nition and interpretation" and 11 out of 21 contributions were published in Fresenius J Anal Chem (Fresenius and Malissa [1992]; Cammann [1992]; Valcarcel [1992]; Zuckerman [1992]; Zhou Nan [1992]; Koch [1992]; Perez-Bustamante [1992]; Ortner [1992]; Danzer [1992]; Green [1992]; Stulik and Zyka [1992]; Kuznetsov [1992]). The ˇrst prize winner { Cammann [1992] { deˇned analytical chemistry \as the self-reliant chemical sub-discipline which develops and delivers appropriate methods and tools to gain information on the composition and structure of matter, especially concerning type, number, energetic state and geometrical arrangement of atoms and molecules in general or within any given sample volume. ...In analytical chemistry, special techniques are used to transform measured chemical signals, derived mostly from speciˇc interaction between matter and energy, into information and ordered knowledge". Other remarkable aspects expressed in the published contributions of the competition are the following: Analytical chemistry . . .

\...isthe chemical metrological science" (Valcarcel [1992]) \. . . can also be considered applied physical chemistry"1 (Cammann [1992]) \...is a science devoted to the analytical cognition of substances: their properties, composition, structure and state, steric and inner relations, behaviour in chemical reaction systems at all levels of the chemical organisation of matter" (Zuckerman [1992]) \...is a branch of science which comprises the theory and practice of acquiring information about chemical characteristics of any matter of system, present in particular state, from its bulk or from a speciˇed region ..." (Zhou Nan [1992]) \...isamultidiscipline, comprising various ˇelds of chemistry with special understanding of physics, mathematics, computer science, and engineering" (Koch [1992]) \. . . uses chemical, physicochemical, and physical or even biological methods for analytical signal production, followed by problem- and matter- related signal processing and signal interpretation in order to provide

1 Note of the author: this interpretation is not reversible: physical chemistry is not theoretical analytical chemistry 4 1 Object of Analytical Chemistry reliable (quality assured) qualitative, quantitative and/or structural information about a sample" (Koch [1992]) \...isthe branch of chemistry, the aim of which is the handling of any type of matter in order to separate, identify, quantify and speciate its components by extracting the pertinent information of analytical interest contained in a representative sample" (Perez-Bustamante [1992]) \...is based upon the symbiotic knowledge and application of chemistry, physics and applied mathematics, to establish a congruent trinomium involving: chemical species { instrumentation and metrology { data handling and processing (chemometrics)" (Perez-Bustamante [1992]) \...isthe science of the creative derivation of information using proper methods to answer the following four basic questions which also resemble the relevant fundamental ˇelds of analytical chemistry: What and how much? (Bulk analysis), How structured? (Structure analysis), How bound? (Speciation), How distributed? (Topochemical analysis)" (Ortner [1992]) \. . . embraces that domain of the physical sciences that allows the interaction of molecules and matter to be understood and their composition to be determined." (Green [1992]) All these deˇnitions express essential aspects of analytical chemistry and the analytical work. Some others { with originality { could be added, such as that from Murray [1991] who characterized analytical chemistry briey and aptly as \the science of chemical measurements". Different opinions can be found about the status of analytical chemistry as being a branch of chemistry independent from other chemical disciplines or being a physical discipline (Green [1992]), or even being an autonomous science, occasionally called Analytics or Analytical Sciences. On the other hand, wide agreement can be stated about the aim to obtain information on matter via representative samples and the inclusion of structural information. Remarks on the general importance of analytical signals can be repeatedly found. Considering the recent development of analytical chemistry and the sig- niˇcance of analytical signals for which reasons will be given in Chap. 3, the following object characterization is proposed:

Analytical Chemistry is the science of chemical measurement. Its object is the generation, treatment and evaluation of signals from which information is obtained on the composition and structure of matter.

Principle goals are the identiˇcation and recognition of sample constituents with regard to their type (elements, isotopes, ions, modiˇcations, complexes, functional groups, compounds, viz. simple, macromolecular or biomolecular, and occasionally sum parameters, too), their amount (absolute or relative: content, concentration), and binding state (oxidation state, binding form and type: inorganic, organic, or complex bound). 1.2 Repertoire of Analytical Chemistry 5

Fig. 1.2. The analytical trinity; according to Betteridge [1976]

The opinions about the status of analytical chemistry { being an independent or an auxiliary science { have varied over many decades. However, it is not really a desirable to be independent among the scientiˇc or chemical disciplines and is not a disadvantage to be thought of as an auxiliary science. In some respects, various respectable sciences like mathematics, physics, and biology have a service function, too. Therefore, the question about independence is relative and of secondary importance. In contrast to classical analysis, the concept of modern analytical chemistry has changed in so far as the problem that has to be solved is included in the analytical process. The analytical chemist is considered as a \problem solver" (Lucchesi [1980]) and the concept is represented in the form of the \analytical trinity" (Betteridge [1976]) as shown in Fig. 1.2. Nowadays, analytical chemistry has a large variety of methods, techniques and apparatus at its disposal and is able to play its \instruments" with high virtuosity. Therefore, the wide range of performance which analytical chemistry can achieve is extremely varied and extends from simple binary decisions (qualitative analysis) to quantitative analysis at the ultra- trace level, from structure elucidation and species identiˇcation to studies of the dynamics and the topology of multispecies systems by means of temporally and spatially high-resolving techniques.

1.2 Repertoire of Analytical Chemistry

The continual progress of analytical chemistry is attributed to the increasing demands from science and technology as well as from society. Performance parameters have rapidly and drastically improved by the high demands of the main focuses of development. In Fig. 1.3 it is shown how the efˇciency of analytical methods has successively improved in the last 50 years. The progress was caused by the development of completely new methods, techniques, and principles (e.g. microprobe techniques) as well as the introduction of new system components into common instruments (e.g. ash 6 1 Object of Analytical Chemistry

Fig. 1.3. Technical progress in analytical chemistry with regard to sample mass m, limit of detection LD, sample areas (cross-section) l, and time expenditure/time resolution t

excitation, capillary columns) and methods (e.g. enrichment techniques, hyphenated methods). Additionally, the use of chemometric procedures has considerably improved the power and range of analytical methods. Analytical investigations usually concern samples which are temporally and locally invariant. This kind of analysis is denoted as bulk analysis (average analysis). On the other hand, analytical investigations can particularly be directed to characterize temporal or local dependences of the composition or structure of samples. One has to perform dynamic analysis or process analysis on the one hand and distribution analysis, local analysis, micro analysis, and nano analysis on the other. According to the demands of the analysis, analytical chemistry can be classiˇed into analysis of major components (major component analysis, precision analysis, investigation of stoichiometry), minor components, and trace components (trace analysis, ultra trace analysis). On the other hand, analytical problems are differentiated according to the number of analytes involved. Accordingly, single component and multicomponent analysis are distinguished. 1.2 Repertoire of Analytical Chemistry 7

Fig. 1.4. The tetrahedron of the analytical repertoire

In principle, structure analysis can be considered as distribution analysis in atomic dimensions. However, from the practical point of view it makes sense to deal separately with structure analysis and to differentiate between molecular structure analysis and crystal structure analysis. Further structure investigations concern near-orders in solids and liquids (e.g. glass). The relations between the questions that are answered by analytical chemistry are shown in Fig. 1.4. The tetrahedron represents the basic analytical repertoire in a simpliˇed way. It can be seen that all the analytical treatments are connected with each other. In Fig. 1.5 the repertoire of analytical chemistry is classiˇed more in detail according to element and structure analysis and according to the extent of quantiˇcation. The different ways of species analysis { qualitative and quantitative { are well known. However, in structure analysis, they can also be differentiated between qualitative and quantitative ways according to the type and amount of information obtained (Eckschlager and Danzer [1994]). Identiˇcation of a sample or a given constituent may have an intermediate position between species and structure analysis. In any case, identiˇcation is not the same as qualitative analysis. The latter is the process of determining if a particular analyte is present in a sample (Prichard et al. [2001]). Qualitative analysis seeks to answer the question of whether certain components are present in a sample or not. On the other hand, identiˇcation is the process of ˇnding out what unknown substance(s) is or are present (Eckschlager and Danzer [1994]). In Sects 9.1 and 9.3 it will be shown that there is a 8 1 Object of Analytical Chemistry

Fig. 1.5. Survey of element and structure analytical standard procedures clear difference between the information contents of qualitative analysis and identiˇcation. Another aspect of modern analytical chemistry is the possibility of multicomponent analysis. Especially spectroscopic and chromatographic methods are able to detect and determine a large number of species simultaneously. Therefore, such methods like ICP-OES, ICP-MS, TXRF, and chromatography are the work-horses in today's analytical chemistry. Notwithstanding the formal classiˇcation given in Fig. 1.5 there is no fundamental difference between qualitative and quantitative analysis. In each case a speciˇc signal is generated which may be evaluated to meet any component of the following logical sequence: Detection ! Assurance ! Semiquantifying ! Quantifying This chain of information can be broken and applied at each point according to the special demands. By the way, even the detection limit involves a numerical estimation. A rough gradation of analyte amounts has been done by de Gramont [1922] who investigated 82 elements in minerals, ores and alloys by means of atomic spectroscopy using so-called \raies ultimes" (last lines, ultimate lines, i.e. such lines which disappear at deˇnite concentrations). The application of combinatorial principles in chemical synthesis, particularly in the search for active substances, requires analytical methods with high throughput (von dem Bussche-Hunnefeld et al. [1997]). Screening techniques can be used to analyse a large number of test samples in a short 1.2 Repertoire of Analytical Chemistry 9

Fig. 1.6. Absolute masses of samples and analytes (a) and relative analyte amounts (b) relevant in analytical investigations time. Methods of screening must be inexpensive with regard to both time and cost. Naturally they must be reliable enough that the risk of erroneous decisions (false positive and false decisions, respectively) is low. Structure elucidation of a compound follows another logical sequence:

Constituents ! Gross composition ! Sum formula ! Constitution ! Conˇguration ! Conformation ! Quantitation: Intramolecular Dimen- sions

This strengthens the case for treating structure analysis as a particular ˇeld of analytical chemistry despite the fact that, from the philosophical point of view, structure analysis can be considered as distribution analysis (topochemical analysis) of species in atomic dimensions. Structure analysis of solids follows a similar scheme like that given above. The characteristics of molecules are then linked with those of crystals and elementary cells. A quantiˇcation of the repertoire of analytical chemistry is shown in Fig. 1.6. The ˇeld of operation covers over 30 orders of magnitude and more when the amount of lots is included. On the other hand, the relative amounts (contents, concentrations) with which the analyst has to do covers 20 orders and more because single atom detection has become reality now. The ˇelds of application of analytical chemistry extend from research to service, diagnosis, and process control, from science to technology and society, from chemistry to biology, health services, production, environmental protection, criminalogy and law as well as from chemical synthesis to materials sciences and engineering, microelectronics, and space ight. In brief, analytical chemistry plays an important role in every ˇeld of our life. 10 References Today, analytical chemistry has such a wide variety of methods and techniques at its disposal that the search for general fundamentals seems to be very difˇcult. But independent from the concrete chemical, physical and technical basis on which analytical methods work, all the methods do have one principle in common, namely the extraction of information from samples by the generation, processing, calibration, and evaluation of signals according to the logical steps of the analytical process.

References

Betteridge D (1976) Tell me the old old story or the analytical trinity. Anal Chem 48:1034A Cammann, K (1992) Analytical chemistry { today's deˇnition and interpretation. Frese- nius J Anal Chem 343:812 Danzer, K (1992) Analytical chemistry { today's deˇnition and interpretation. Fresenius J Anal Chem 343:827 Danzer, K, Than, E, Molch, D (1976) Analytik { Systematischer Uberblick . Akademische Verlagsgesellschaft Geest & Portig, Leipzig de Gramont, MA (1922) Tableau des raies de grande sensibilite des el ements destine aux recherches analytiques. C R Hebd Seances Acad Sci 171:1106 Eckschlager, K, Danzer, K (1994) Information theory in analytical chemistry. Wiley, New York, Chichester, Brisbane Fresenius, W, Malissa, H (1992) Competition \analytical chemistry { today's deˇnition and interpretation\. Fresenius J Anal Chem 343:809 Green, JD (1992) Analytical chemistry { today's deˇnition and interpretation. Fresenius Z Anal Chem 343:829 Kellner, R (1994) Education of analytical chemists in Europe. Anal Chem 66:98A Kellner, R, Mermet, J-M, Otto, M, Widmer, HM [eds] (1998) Analytical chemistry. The approved text to the FECS curriculum analytical chemistry. Wiley-VCH, Weinheim, New York, Chichester Koch, KH (1992) Analytical chemistry { today's deˇnition and interpretation. Fresenius J Anal Chem 343:821 Kusnetsov, VI (1992) Analytical chemistry { today's deˇnition and interpretation. Frese- nius J Anal Chem 343:834 Lewenstam, A, Zytkow, JM (1987) Is analytical chemistry an autonomous ˇeld of science? Fresenius Z Anal Chem 326:308 Lucchesi, CA (1980) The analytical chemist as problem solver. Internat Lab 11/12:67 Malissa, H (1987) Analytical chemistry: kitchenmaid or lady of science { deduction versus induction. Fresenius Z Anal Chem 326:324 Murray, RW (1991) Analytical chemistry: the science of chemical measurement. Anal Chem 63: 271A Ortner, HM (1992) Analytical chemistry { today's deˇnition and interpretation. Fresenius J Anal Chem 343:825 References 11

Oswald, W (1894) Die wissenschaftlichen Grundlagen der Analytischen Chemie. Verlag Wilhelm Engelmann, Leipzig Perez-Bustamante, JA (1992) Analytical chemistry { today's deˇnition and interpretation. Fresenius J Anal Chem 343:823 Prichard, E, Green, J, Houlgate, P, Miller, J, Newman, E, Phillips, G, Rowley, A (2001) Analytical measurement terminology { handbook of terms used in quality assurance of analytical measurement. LGC, Teddington, Royal Society of Chemistry, Cambridge Reilley, CN (1965) AS National Meeting, Fisher Award Address Stulik, K, Zyka, J (1992) Analytical chemistry { today's deˇnition and interpretation. Fresenius J Anal Chem 343:832 Szabadvary, F (1966) History of analytical chemistry. Pergamon Press, Oxford Valcarcel, M (1992) Analytical chemistry { today's deˇnition and interpretation. Fresenius J Anal Chem 343:814 Veress, GE, Vass, I, Pungor, E (1987) Analytical chemical methods as systems producing chemical information by inference. Fresenius Z Anal Chem 326:317 von dem Bussche-Hunnefeld C, Balkenhohl F, Lansky A, Zechel C (1997) Combinatorial chemistry. Fresenius J Anal Chem 359:3 WPAC (1993) Edinburgh deˇnition of analytical chemistry; see Kellner, R (1994) Yordanov, N (1987) The position of analytical chemistry from the viewpoint of moderate scientism { a critical outlook. Fresenius Z Anal Chem 326:303 Zhou Nan (1992) Analytical chemistry { today's deˇnition and interpretation. Fresenius J Anal Chem 343:812 Zolotov YA (1984) Khimiczeskaja enciklopedia. Nauka, Moscow Zuckerman, AM (1992) Analytical chemistry { today's deˇnition and interpretation. Fre- senius J Anal Chem 343:817 2 The Analytical Process

Analytical chemistry is a problem-solving science. Independent from the concrete analytical method, the course of action, called analytical process,is always very similar. The analytical process starts with the analytical question on the subject of investigation and forms a closed chain to the answer to the problem. Using a proper sampling technique a test sample is taken that is adequately prepared and then measured. The measured data are evaluated on the basis of a correct calibration and then interpreted with regard to the object under study. The analytical process in the broader sense is represented in Fig. 2.1. Frequently { in a stricter sense { only the lower part (grey background) from sample through to information (or essential parts from it) are regarded as representing the analytical process. The lower part of the analytical process (grey) is { mostly merged as a black box { sometimes known as the chemical measurement process (CMP); see Currie [1985, 1995, 1999].

Problem Sampling strategy Problem solving

Object under study

Sampling Interpretation

Test Measuring Measured Analytical 1 2 Signal 3 4 5 Information sample sample values data

6 Latent 7 variables

1 Sample preparation 4 Evaluation/calibration 2 Measurement 5 Data evaluation & interpretation 3 Signal validation 6 & 7 Chemometrics

Fig. 2.1. The analytical process (Danzer [2004]) 14 2 The Analytical Process

Fig. 2.2. General principle of processing information

Different from former representations of the analytical process (Danzer et al. [1976], Lucchesi [1980], Tyson [1989], Kellner et al. [1998]), signal generation is inserted as a separate step. This is because at ˇrst signals are produced from measuring samples and then measured values are taken from the signals. For this, signals must be selected from a signal function. After validation, parameters are chosen, the values of which are subject to evaluation (e.g., signal position, differences of positions, signal intensity measured in form of peak height or peak area). In addition to the measured values and the analytical values (e.g. content, concentration), latent variables are included in the scheme. Latent variables can be obtained from measured values or from analytical values by means of mathematical operations (e.g. addition, subtraction, eigenanalysis). By means of latent variables and their typical pattern (represented in chemometric displays) special information can be obtained, e.g. on quality, genuineness, authenticity, homogeneity, origin of products, and health of patients. The analytical process in the stricter sense or chemical measurement process, respectively, has a conspicuous similarity with the general information process which is shown in Fig. 2.2. Interference plays an analogous role in the course of the analytical process as well as noise, which is mainly manifested by random deviations. The scheme given in Fig. 2.1 represents normal analytical procedures in off-line analysis. It contains all the steps that must be considered in principle. However, there could be reasons to reduce the course of action. Not in all cases is the analyst able to take samples by himself or check the sampling procedure. Sometimes he or she must accept a situation in which he or she has to receive a given sample (e.g., in extreme cases, extraterrestrial samples, autopsy matter). In some cases the object under study has to be continuously monitored. Then on-line analytical methods are applied by which the system can be directly measured. The analytical process then runs without sampling and sample preparation, as can be seen in Fig. 2.3a. The analytical process is shortened even more in the case of in-line analysis where measurement and 2.1 Principles of Sampling 15

Fig. 2.3. The analytical process in case of on-line analysis (a) and in-line analysis (b) control equipment are coupled with and installed in the object under study (Fig. 2.3b). Other forms of analytical investigations may also be applied, such as at- line analysis (discontinuous direct analysis of objects, e.g., noninvasive blood glucose monitoring) and trans-line analysis (remote sensing, teleanalysis). In the following, the stages of the analytical process will be dealt with in some detail, viz. sampling principles, sample preparation, principles of analytical measurement, and analytical evaluation. Because of their signif- icance, the stages signal generation, calibration, statistical evaluation, and data interpretation will be treated in separate chapters.

2.1 Principles of Sampling

Sampling is the most important step in the course of the analytical process. It is essential for obtaining reliable analytical results that the samples are taken from the lot in a representative way. Therefore, sampling strategy and design, sampling technique and sample size must be harmonized in an optimum way. There exist general guidelines for sampling (Horwitz [1990], Gy [1982, 1992], Kraft [1980], Wegscheider [1996]) as well as special instructions for various ˇelds of application (e.g. Taggart [1945], Keith [1988], Markert [1994] Barnard [1995]). The general demand on sampling is that it has to be carried out in a representative way, to be precise, representative with regard to both the properties of the material and the analytical problem. This twofold representativeness of the sample means that sampling strategies are different in case of average analysis (bulk analysis) and testing homogeneity within a lot of n batches. Figure 2.4 shows two possible sampling schemes for answering both questions. Sampling is a crucial step of the analytical process, particularly in cases where there are large differences between the material under investigation and the test sample (laboratory sample) with regard to both amounts and properties, especially grain size, uctuations of quality and inhomogeneities. 16 2 The Analytical Process

a b

Lot, batch n units ...

p primary samples ... Gross samples ......

Composite sample ...... q subsamples of 1st degree

...... q subsamples of 2nd degree ...... (3rd degree) ... q⋅ r test samples

test portions (q⋅ r)

Fig. 2.4. Sampling schemes for bulk analysis (a) and for testing homogeneity of several batches (b), according to Danzer [1995a] and following Horwitz [1990]

A material is regarded as being inhomogeneous if the chemical composition is not the same in all of its parts, i.e. { In spatial dimensions or in partial volumes of compact solids, or { In portions of bulk materials (grained and powdered substances) and { In time sequences of natural or technical processes. The technical terms homogeneity and inhomogeneity deˇned in analytical chemistry must be distinguished from the physicochemical concept of homogeneity and heterogeneity (Danzer and Ehrlich [1984]). Whereas the thermodynamical deˇnition refers to morphology and takes one-phase- or multi-phase states of matter as the criterion, the analytical-chemical def- inition is based on the concentration function x = f (l) (2.1) 2.1 Principles of Sampling 17

Fig. 2.5. Illustration of the terms homogeneity, heterogeneity and inhomogeneity from the physicochemical and analytical viewpoint

and is statistically determined. Instead of the distance l applied in Eq. (2.1), other spatial dimensions of the sample (area, surface or volume, respectively) or even the time t may be of relevance. In contrast to the physicochemical categorical properties of homogeneity and heterogeneity, the transition from homogeneity to inhomogeneity in analytical chemistry (the degree of homogeneity) may be characterized by a variable, see, e.g., Eq. (2.9). Figure 2.5 illustrates the state of affairs, and shows that heterogeneous material may be characterized by an inhomogeneous (C) or homogeneous (D) concentration function dependent on the relation between the total variation of concentration and the uncertainty of measurement on the one hand and the sample amount (or microprobe diameter in case of distribution- analytical investigations) on the other. The sample A is homogeneous both from the physicochemical and the analytical-chemical point of view because the variation of the concentration is within the uncertainty of the analytical measurement unc(x). In contrast, sample B is homogeneous from the physico-chemical but not from the analytical-chemical viewpoint because the systematical change of the concentration exceeds unc(x). Whereas sample C is heterogeneous from the physico-chemical point of view and inhomogenous from the analytical- chemical viewpoint (the concentration deviations plainly exceed unc(x)), sample D is heterogeneous from the physico-chemical standpoint. The an- 18 2 The Analytical Process

a bc d

llocal

x l

The local frequency 1/llocal is practical zero low medium high (constant concentration) (concentration gradient) (periodical fluctuation) (stochastic fluctuation)

Fig. 2.6. Characteristic concentration functions and local frequencies alytical decision about homogeneity depends on the diameter of the microprobe or the size of the portion of the test sample. If the sample size is large enough the material may be characterized to be homogeneous as illustrated in Fig. 2.5D. In analogy to the time frequency, f , the spatial concentration behaviour may be characterized by the local frequency, flocal =1=llocal. In Fig. 2.6, four types of spatial concentration functions are shown. These types and combinations of them can characterize all the variations of concentration in analytical practice both in one- and more-dimensional cases. In addition to the local frequency, the amplitude of the concentration uctuation is a crucial parameter for the characterization of inhomogeneity (see Fig. 2.5, right hand side). It is essential for sampling problems that inhomogeneities, as real concentration differences x between parts of sample portions or areas, respectively, can only be ensured if x is signiˇcantly greater than the uncertainty U (x) of the analytical method. The criterion for testing homogeneity is the F -test: 2 total F = 2 (2.2) anal which takes as basis the addition of variances according to the model 2 2 2 total = anal + inhom : (2.3a) 2 2 The inhomogeneity variance inhom determines the sampling error samp which can be estimated experimentally and which is crucial for representative sampling1 2 2 2 total = anal + samp (2.3b) The relationship between the estimated variances s2 has to consider the number of primary samples, p , subsamples, q, and test samples, q r

1 The inhomogeneity- and the sampling variance are adequate only with a given risk 2 ˛ 2 of error ˛: inhom = samp 2.1 Principles of Sampling 19

2 2 s ssamp s2 = anal + : (2.4) x q r p q In the case of sampling according to the scheme given in Fig. 2.4a, the critical number of samples required at least for reliable analytical results can be derived from Eq. (2.4) (Danzer [1995b]): 2 2 psanal + rssamp q = 2 (2.5) rpsx General relationships such as Eq. (2.6) which has been created by Kraft [1980]: t s 2 q = (2.6) U lack information what s and U precisely mean (U is described as the precision of sampling plus analysis and relates to the combined uncertainty as deˇned in Sect. 4.2 according to ISO [1993] in certain respect. The critical weight of samples can be derived from the general condition of representativeness of sampling expressed by the null hypothesis H0: 2 2 Fisher total = anal which is tested by means of 's F -test s2 ^ total F = 2 : (2.7) sanal The general principles of testing chemical homogeneity of solids are given e.g. by Malissa [1973], Cochran [1977], and Danzer et al. [1979]. 2 2 The terms of variation total and anal can be separated by analysis of variance (Sect. 5.1.1). According to Danzer and Kuchler [1977] there exists an exponential dependence between the total variance and the reciprocal sample mass a s2 = s2 exp (2.8a) total anal m where a is material-speciˇc parameter. Developing Eq. (2.8a) into an arith- metical progression, the following relationship results: a s2 = s2 1+ (2.8b) total anal m when the higher powers of m are neglected. The exponentially decrease of the total variance with increasing sample mass is shown in Fig. 2.7. It can be seen that the uncertainty of sam- 2 pling, ssamp, decreases and becomes statistically insigniˇcant when the sample amount m exceeds the critical sample mass. Instead of mcrit the proportional critical sample volume vcrit may also be considered, represented, e.g. by a critical microprobe diameter dcrit. Results of homogeneity investigations of alloys, ores, and lamellar eutectics by EPMA (Electron Microprobe Analysis), which correspond to the curve of Fig. 2.7, have been presented by Danzer and Kuchler [1977]. 20 2 The Analytical Process

Fig. 2.7. Dependence of the total variance on the sample amount characterized by mass; mcrit is the critical sample mass. The statement of homogeneity or inhomogeneity is always related to a given analyte A

In the course of time, several measures have been proposed to characterize homogeneity of solids, such as homogeneity constants, factors etc. (Gy [1982], Singer and Danzer [1984], Kurfurst [1991], Ingamells and Switzer [1973], Ingamells [1974, 1976]). However, unfortunately, quantities like this have a special background with regard to the material and, therefore, do not have any general decision power. Taking the homogeneity number of Singer and Danzer [1984] as starting point, a useful measure of homogeneity can be developed. The state of affairs with regard to the distribution of a particular species A in bulk material can be characterized by the quantity homogeneity deˇned by F^ hom(A)=1 : (2.9) F1˛;1;2 This homogeneity value will become positive when the null hypothesis is Fisher ^ not rejected by 's F -test (F F1˛;1;2 ) and will be closer to 1 the more homogeneous the material is. If inhomogeneity is statistically proved ^ by the test statistic F>F1˛;1;2 , the homogeneity value becomes negative. ^ In the limiting case F = F1˛;1;2 , hom(A) becomes zero. In the case that inhomogeneities are not attributed to random variations but to systematic changes as, e.g. shown in Fig. 2.5B, then other strategies of sampling and evaluation must be used. Numerous different sampling designs have been proposed. Surveys can be found, e.g. in Cochran [1977], Gilbert [1987], Keith [1988] Nothbaum et al. [1994], and Einax et al. 2.1 Principles of Sampling 21 [1997]. Diverse systematical strategies of sampling require other statistical evaluation procedures than simple analysis of variance (ANOVA) as applied in case of stochastic sampling. In case of regular arrangements of sampling points, the statistical evaluation can be carried out in several ways (Danzer [1995b], Einax et al. [1997]): (1) Two-way layout ANOVA according to Danzer and Marx [1979]. In this way it is possible to obtain not only information on homogeneity/inhomogeneity, but also on certain preferred directions of inhomogeneities. (2) Regression analysis by means of linear or quasi-linear models (Parczewski [1981], Singer and Danzer [1984], Parczewski et al. [1986]). (3) Gradient analysis (Parczewski [1981], Singer and Danzer [1984], Parczewski et al [1986], Parczewski and Danzer [1993]) which is based on two-dimensional regression models and adds pictorial information to statistical decisions. Figure 3.12 shows such a graphical representation of an element distribution on a surface. (4) Pattern recognition methods (PARC) can be used if no a priori information on the type of inhomogeneities (stochastic or systematic) is available. By reversing the aim of PARC (classifying objects into classes), homogeneity is proved by the impossibility to form point classes having signiˇcantly different concentrations (Danzer and Singer [1985]). (5) Trigonometric functions have been used by Inczedy [1982] to describe inhomogeneities quantitatively. (6) Autocorrelation and time series analysis have been successfully applied in testing spatial inhomogeneities (Ehrlich and Kluge [1989], Do- erffel et al. [1990]). This techniques are generalized in the theory of stochastic processes (Bohacek [1977a, b]) which is widely used in chemical process analysis and about them. The main goal of time-series analysis (Box and Jenkins [1976], Chat- field [1989], Metzler and Nickel [1986]) apart from process analysis is time-dependent sampling. In both cases uctuations in time x(t) matter and can be considered as a simple stochastic process or as time series. Process sampling and analysis will be carried out with several aims (Kateman [1990]): { Control, i.e. sampling, analysis and reconstruction of the process { Monitoring for process safety and warnings { Description, i.e. mathematical decomposition of the variability by time series analysis A time series x(t) can be decomposed according to Hartung et al. [1991] into even (smooth) components, xeven(t), seasonal components, xseas(t), and irregular components, xirreg(t): 22 2 The Analytical Process

x(t)=xeven(t)+xseas(t)+xirreg(t) : (2.10) Seasonal process components are of particular interest for environmental studies and production processes. The different components of x(t) can be estimated by smoothing and ˇltering methods. The correlation of values within a time series plays an important role in sampling. It can be characterized by means of the autocorrelation function (Doerffel and Wundrack [1986], Chatfield [1989], Hartung et al. [1991])

+T 1 xx() = lim x(t) x(t + )dt (2.11) T !1 2T T where x(t) is the value of the time series at the time t, the time lag (time difference between two values), x(t + ) the value for lag in relation to t, and T the total time. The autocorrelation function corresponds to the time average x(t) x(t + ) of the function x(t). For stochastical time series the autocorrelation function (ACF) is represented by a typical curve as shown in Fig. 2.8. The correlation time (time constant) corresponds to the lag at which the maximum value xx(0) is decreased by the factor 1/e and the correlation becomes practically zero. Correlation time is an important quantity in sam-

Fig. 2.8. Autocorrelation function (ACF) of a stochastic process. c is the correlation time (time constant) 2.2 Sample Preparation 23 pling because measured values x(t) and x(t + ) at intervals >c can be regarded as independent from each other. Similar models have been applied in geological exploration and environmental studies. There semivariogram analysis (Akin and Siemens [1988], Einax et al. [1997]) plays a comparable role than autocorrelation analysis for the characterization of stochastic processes. Representative sampling is the ˇrst and most important prerequisite to produce accurate and precise analytical results. The fact that sampling may be a crucial step with regard to reliable chemical measurements has been well-known for years and has promoted the development of advanced practical methods as well as the forming of theoretical fundamentals of sampling. Excellent reviews are given by Kraft [1980], Kratochvil et al. [1984], Gy [1992], Kateman [1990], and Kateman and Buydens [1993]. Nomenclature for sampling in analytical chemistry is given in Kratochvil et al. [1984] and Horwitz [1990].

2.2 Sample Preparation

Only in a few cases are test samples measurable without any treatment. As a rule, test samples have to be transformed into a measurable form that optimally corresponds to the demands of the measuring technique. There- fore, sample preparation is a procedure that converts a test sample into a measuring sample. Whereas test samples represent the material in its original form, measuring samples embodies a form that is able to interact with the measuring system in an optimum way. In this sense, measuring samples can be solutions, extracts, pellets, and melt-down samples, but also deˇnite surface layers and volumes in case of micro- and nanoprobe techniques. As an adaptation procedure, sample preparation has to fulˇll at least one of the following aims: (1) Making the test sample physically or chemically measurable by the analytical technique (2) Elimination of interferences (3) Improving the relation between the amounts of analyte and matrix Measuring samples are considered as being composed of the analyte (Prichard et al. [2001]) and the matrix. The term \matrix" summa- rizes all the sample constituents apart from the analyte. The relationships between the amounts of sample, analyte, and matrix are given by: { The sample mass

msample = manalyte + mmatrix (2.12a) { The absolute mass of the analyte

manalyte = msample xanalyte (2.12b) 24 2 The Analytical Process { The relative mass of the analyte (content, concentration)

manalyte manalyte xanalyte = = (2.12c) msample manalyte + mmatrix Sample preparation is directed to the conversion of test samples in a physically and chemically measurable form. The measuring sample can require a deˇnite state (gaseous, liquid, or solid) or form (aqueous or organic solution, melt-down tests, and pellets). In other cases, measuring samples have to become diluted or enriched to get an optimum concentration range. It may also be necessary to remove interfering matrix constituents which dis- turb the determination of the analyte.

Physical sample pretreatment. Almost all test samples, including those which are measured by so-called \direct techniques", need physical pretreatment in any form. The most applied physical techniques are: { Change in the physical state: freezing (freeze-drying), crystallizing, con- densation, melting, pressure-pelletizing, vaporization etc. { Improvement of the form: grinding and homogenizing of solid and coarse materials { Surface state: polishing, sputtering or etching of rough surfaces of specimen { Preparation of electrically conductive mixtures and coatings on surfaces, respectively

Chemical sample preparation. The most important chemical and physicochemical procedures of sample pretreatment are: { Dissolving the test sample by water, other solvents, acids, bases, melts, and gases { Extraction of a group of analytes from complex matrices by liquid extraction (in recent time increasingly by SPME) { Separation of interfering constituents to make the analyte determination possible or more speciˇc (in some cases masking serves the same purpose) { Enrichment of the analyte to improve the signal-to noise-ratio and the detection limit of the analytical method, respectively { Transforming the analyte in a chemical measurable form by means of chemical reaction (e.g. redox reaction, complex forming) In some cases, sample preparation techniques are linked directly with analysing techniques using so-called multi-stage combined systems (Tolg [1979]) which frequently work in a closed system. The principle is shown in Fig. 2.9 in contrast to a direct analytical system. Separation technique is a particularly wide ˇeld in sample pretreatment. Some essential methods of analytical determination are based on separa- 2.2 Sample Preparation 25

Fig. 2.9. Comparison of analytical multi-stage techniques (left) and direct instrumental techniques (right) tion principles, particularly mass spectrometry and chromatographic techniques. In recent times these methods have been coupled amongst themselves and with other methods to achieve separations of high performance. Also other techniques of sample preparation are directly coupled with detection units. Today, such combined methods are called hyphenated techniques (Hirschfeld [1980]) and play an important role in modern analytical chemistry. In contrast to combined systems, hyphenated techniques consist of two or more analytical systems each of which is independently applicable as an analytical technique. Usually, the connection is realized by means of an interface and the system is controlled by a computer. With regard to integrated sample treatment, separation and transfer, hyphenated methods like GC-MS, HPLC-MS, GC-IR, GC-IR-MS, GC-AAS, GC-ICP-MS, MS-MS, and 26 2 The Analytical Process

Fig. 2.10. Composition of a test sample of wine (schematic representation in the middle; the size of each area symbolizes the relative amount of the respective component or sort of constituents) and of the measuring samples of inorganic and organic components, respectively (left and right diagrams)

MSn are of particular interest. MSn stands for the n-fold coupling of mass spectrometers, alternatively serving as separation and detection instrument. By hyphenated techniques the dimensionality of analytical information (see Sect. 3.4) and, therefore, also the information amount (see Sect. 9.3) is sig- niˇcantly increased (Eckschlager and Danzer [1994]). Sometimes it is necessary to apply two (or more) variants of sample preparation to get different measuring samples from only one test sample. This is the case if various problems have to be solved, e.g. determination of major- and ultra trace constituents, comparison of depth- and surface proˇles, or analysis of inorganic and organic trace components in the same test sample. An example is shown in Fig. 2.10 where in a test sample of wine both inorganic and organic trace constituents have to be determined and, therefore, different measuring samples must be prepared.

2.3 Principles of Analytical Measurement

Measurement always involves comparison of an object with a suitable standard. Whereas physical elementary quantities like length and mass can be compared directly, chemical quantities are mostly compared indirectly. It 2.3 Principles of Analytical Measurement 27

Fig. 2.11. Elastic and inelastic interactions between energy and measuring sample is the goal of analytical measurements to make available the information on the composition and structure, respectively, which is contained in the sample in a latent way. This hidden (static) information must be released from the measuring sample by chemical reactions or energetic interactions by which real (dynamic) signals are generated. The interactions to which the measuring samples are exposed can be of elastic or inelastic type (in the physical sense). Figure 2.11 shows schematically the difference between elastic and inelastic interactions. Elastic interactions are characterized by the condition that no energy is exchanged between the measurement system and the measured sample:

E = Ein Eout = 0 (2.13)

In other words, Eout = Ein. The energy (electromagnetic radiation or particle beam) is neither absorbed nor otherwise energetically changed while coming into contact with or passing through the measuring sample. What happens instead of this is a change in the spatial structure of the radiation by re- fraction, reection, diffraction or scattering. From such structural changes of the energy system Eout information can be obtained on the structural arrangement of the constituents in the measuring sample. The majority of analytical methods are based on inelastic interactions. In contrast to elastic interactions, in case of inelastic interactions both the energy of the measuring system, Ein, and the intrinsic energy of the constituents of the measuring sample, E0, will be changed in such a way that the following condition is fulˇlled: 28 2 The Analytical Process Ekin = (Ein Eout)= U = (Eexc E0) (2.14) where each change in the kinetic energies Ekin deˇnitely corresponds to changes in the intrinsic energy U of the sample constituents. The amounts of kinetic and intrinsic energy, jEkinj and jU j, characterize the sort of sample constituents whereas the sum of all the energy quanta characterizes the amount of these species. As a result of the interchanges, signals are produced which contain both information on the type of constituents and their amounts. Ekin and U determine the signal position, z, and the intensities of each signal, y , correspond to the amount of the belonging species. The latent information of the measuring sample is transferred via an energetic carrier into analytical information which is manifested by signals. Their parameters correspond to the coding (encoding) process in information systems. For the formal representation of the analytical coding the following analytical quantities are introduced: (a) The categorical quantities (discrete variables) Q = A;B;C;:::;N stand for the chemical entities being investigated (elements, isotopes, ions, modiˇcations, complexes, functional groups, compounds, viz. simple, macromolecular or biomolecular, and occasionally sum parameters). These entities are called analyte (Prichard et al. [2001]). All the components of the sample other than the relevant are put together under the term matrix.

(b) The quantity xQ, which is a continuous variable as a rule, characterizes the amount of the analyte as an absolute or relative measure (e.g. mass, content or concentration). The analyte amount xQ is the original goal of quantitative analysis but is frequently determined indirectly. As the quantity subjected to measurement, xQ is called measurand (Fleming et al. [1997], Prichard et al. [2001]). By the operations coding (analytical measurement) and decoding (analytical evaluation) information will be transformed from the sample domain into the signal domain and vice versa as shown in Fig. 2.12. Therefore, quantities which correspond to that of the sample domain (Q and xQ) must also exist in the signal domain. The characteristics in the signal domain are:

(c) The variables zi which characterize the values of the signal positions. Respective zi-positions correspond to given Qs according to assignment rules and tables and { in rare cases { natural and empirical laws and equations, respectively. Such relationships between zi and Q represent a special type of calibration (\qualitative" or \identity-" calibration).

(d) On the other hand, the variables yzi characterize the signal intensity at given positions zi. The relationship between the intensity of a deˇnite signal and the amount of the belonging constituent Q is rarely known in form of theoretical connections. In most cases the relation between

yzi and xQ is modelled by experimental calibration. 2.3 Principles of Analytical Measurement 29

Fig. 2.12. Relationship between sample domain and signal domain in element analysis (a) and structure analysis (b). The representation in the sample domains is shown in different forms, as a block diagram and a list in case of (a) and as constitution formula and structure matrix, respectively, in case (b)

A classiˇcation of the basis of these analytical quantities with regard to their mutual relationships and dependencies from physical quantities like space co-ordinates and time will be given in Sect. 3.4 according to the dimensionality of analytical information. Coding and decoding are reverse operations. Coding can be imagined to consist of two steps:

{ Coding of the (latent) sample information on Q into energy signals jU jQ { Coding of these energy quantities jU jQ into measurable signals zi (measuring quantities) Both steps are characterized by the corresponding natural laws as well as empirical relations caused by measuring instruments and conditions. All such laws and rules are put together by the measuring function (calibration function). Taking the forms of intrinsic energy as the basis of the measuring principle, analytical methods can be classiˇed according to the scheme given in Fig. 2.13. Another scheme arranged according to the forms of energy exchanged between energy system and measuring sample is shown in Fig. 2.14 in form of the so-called \Benninghoven matrix". Both chemical reactions and physical measurements can leave some \traces" in the measured signals. Chemical contaminations can come from 30 2 The Analytical Process

Fig. 2.13. Analytical methods classiˇed according to the changes of the intrinsic energy of analyte atoms, ions, molecules etc.

Fig. 2.14. Updated \Benninghoven matrix" of analytical methods in the context of interactions between different forms of energy the reaction system, e.g. in the form of residues of a reagent. On the other hand, physical signal functions can contain disturbing primary radiation or \contamination" coming from insufˇciencies of the measuring system, e.g. inhomogeneous distribution of radiation or particle beams, ˇnite slit width of optical systems, and noise of electronic components. Therefore, signal functions y (z) always represent a convolution of the true signal function y (z)true and the characteristic function of the analytical instrument h(z) which characterizes all the insufˇciencies of the measuring system:

y (z)=y (z)true h(z) (2.15) 2.4 Analytical Evaluation 31

Fig. 2.15. Changing of the \true" sample signal by the process of analytical measurement

The mathematical symbol is used for the convolution function which means in detail +1 +1

y (z)=y (z)true h(z)= ytrue(z )h()d = ytrue()h(z )d 1 1 where is the integration variable. Additionally, noise n(z) is added to the signal function:

y (z)+n(z)=y (z)true h(z)+n(z) (2.16) The situation is illustrated in Fig. 2.15 where a signal is shown which has been obtained in the analytical reality, distorted and disˇgured by noise and broadening. All of these effects can be returned to a certain degree by techniques of signal treatment like deconvolution, signal accumulation and smoothing, etc.

2.4 Analytical Evaluation

Evaluation of signals is carried out effectively in a 'twofold manner', namely according to: (i) The analytical evaluation functions x = f 1(y ) and Q = f 1(z) which are the respective reciprocal functions of the measuring functions (calibration functions) y = f (x) and z = f (Q), respectively (see Fig. 2.12) (ii) The uncertainty of the analytical measurement and therefore, the uncertainty of the analytical result Regardless of whether quantitative results of element or structure analysis are the matter of evaluation, analytical results have to be reported always in the form result(x)=mean(x) ˙ unc(x) (2.17) 32 2 The Analytical Process

x y

Q Q-z-calibration x-y-calibration (species identification, (quantitative analysis) qualitative analysis) z

Fig. 2.16. Quasi-four-dimensional representation of the connection between qualitative and quantitative evaluation according to Danzer [1995a], Danzer and Currie [1998]

Fig. 2.17. The four analytical quantities in sample and signal domain and the six fundamental functions between them; the functions Q = f 1(x) and z = f 1(y ) do not make much sense in analytical practice 2.4 Analytical Evaluation 33 see Sect. 8.1, where mean(x) is represented by the arithmetic mean x,asa rule, but can also be given by the median, med(x) see Eq. (4.22) and the geometric mean, see Eq. (4.18). Starting with the relation between the analytical function x = f (Q) and the signal function y = f (z) represented in Fig. 2.12, four quantities x, Q, y , and z have to be related to one another. The situation can be characterized in Fig. 2.16 by a quasi-four-dimensional representation (Danzer [1995a], Danzer and Currie [1998]). The foreground of the representation depicts the relationship between the species Q and their characteristic signals z, while behind that, the relationship between signals z, their intensities y , and the species amounts x is established. Taken together, these relationships establish the composition of the sample (the sample domain) and the signal domain, too. Another illustration of the relationships between the four quantities x, Q, y , and z together with the type and terms of the related functions between them is shown in Fig. 2.17. Whereas the x-y -relationships are represented by continuous functions which are linear in most cases, the Q-z-relationships correspond mainly to discrete functions as schematically shown in Fig. 2.18. In analytical practice, Q-z-evaluation refers to the position of signals on energy scales or scales that are related to energy, such as frequency-,

Fig. 2.18. Schematic representations for discrete functions of Q-z-relationships according to natural or empirical laws (a Moseley type, b Kovats type) and empirical connections (c Colthup type, d atomic spectra type) 34 2 The Analytical Process wavelength- or mass/charge coordinates of spectrometers or time coordinates of chromatograms, respectively. The evaluation function Q = f 1(z) that transfers measured z-values into information on Q may be the reciprocal function of one of the following calibration functions (Danzer [1995a]): (a) Deterministic function on the basis of natural laws:

z = fdet(Q) (2.18a)

e.g. Moseley's law of the dependence of X-ray frequencies on the atomic number (Moseley [1913]) (b) Empirical function

z = femp(Q) (2.18b)

like Kovats' indices of homologous compounds in their dependence on retention data in gas chromatography (Kovats [1958]) (c) Empirical connection

z = emp(Q) (2.18c)

represented by tables and atlases, e.g. by tables of characteristic vibra- tions (Colthup et al. [1975]) and iron-atlases in atomic spectroscopy (Harrison [1939], De Gregorio and Savastano [1972]).

While the relations z = fdet(Q) can be derived on the basis of natural laws, the estimation of an empirical function z = femp(Q) for the purpose of identiˇcation and qualitative analysis is mostly carried out by (linear) least squares to ˇt the observed z-values for a set of pure component standards or a multicomponent standard. On the other hand, empirical relationships z = emp(Q) in the form of tables, atlases and graphs are developed by collection and classiˇcation of experimental results. In general, the evaluation function for quantitative analysis is the inverse of the calibration function y = f (x)(Currie [1995, 1999], IUPAC Orange Book [1987, 1988]): x = f 1(y ) (2.19) provided that the relationship between the measured value y and the analyte amount x has been estimated which is mainly the case in analytical chemistry. However, there are also other types of evaluation procedures, e.g. on the basis of natural laws, depending on the nature of the analytical method. The determination of the amounts of analytes can be based on absolute, relative or reference measurements (Hulanicki [1995]) which are based on equations of the general type y = S x (2.20) 2.4 Analytical Evaluation 35 where S is the sensitivity; see Sect. 7.2. For the three mentioned types of analytical measurements the sensitivity is given by mathematically well- deˇned relations, namely in the case of: (1) Absolute measurements by fundamental quantities like Faraday constant and quotients of atomic and molar masses, respectively (coulometry, electrogravimetry, gravimetry, gas volumetry) (2) Deˇnitive measurements by fundamental quantities complemented by an empirical factor, e.g. titre (titrimetry), as well as by well-known empirical (transferable) constants like molar absorption coefˇcient (spectrophotometry), Nernst factor (potentiometry, ISE), and conductivity at deˇnite dilution (conductometry) (3) Direct reference measurements by the relation of measured value to content (concentration) of a reference material (RM) y S = RM (2.21) xRM

On the other hand, indirect reference measurements are based on empirical calibration functions obtained experimentally and frequently based on linear models

y = a + bx + ey (2.22) where the intercept a corresponds to the experimental blank and the slope b to the sensitivity. In analytical practice, some methods using deˇnitive measurements, in principle, are also calibrated in an experimental way (e.g. spectrophotometry, polarography) to provide reliable estimates of S. Calibration functions corresponding to Eq. (2.22) are not generally transferable over long times and not from one laboratory to another. However in the case of blank-free or blank-corrected relations

y = bx + ey (2.23) methods can be robustly calibrated under ˇxed experimental conditions. Then the experimental sensitivity coefˇcients (sensitivity factors) are transferable over time and between laboratories under standard operating conditions. Because of this transferability, such methods are occasionally called \standard-free". Such standard-free methods (not to confuse with \calibration-free" which characterizes absolute measurements) have been developed, e.g., in the ˇeld of OES (Harvey [1947], Danzer [1976]), SS-MS (Ramendik [1990]), and XFA (Sherman [1955], Fei He and van Espen [1991]) for semi-quantitative multielement analysis. For evaluation of multisignal measurements, e.g. in OES and MS, more than one signal per analyte can be evaluated simultaneously by means of multiple and multivariate calibration. The fundamentals of experimental calibration and the relating models are given in Chap. 6. 36 2 The Analytical Process In principle, all measurements are subject to random scattering. Addi- tionally measurements can be affected by systematic deviations. Therefore, the uncertainty of each measurement and measured result has to be evaluated with regard to the aim of the analytical investigation. The uncertainty of a ˇnal analytical result is composed of the uncertainties of all the steps of the analytical process and is expressed either in the way of classical statistics by the addition of variances 2 2 2 2 2 total = sampling + preparation + measurement + evaluation (2.24) or, increasingly in recent time, by means of the empirically calculated combined uncertainty. The estimation of the uncertainty of analytical results is given in Sects. 4.2 and 7.1. Today, analytical evaluation is done on a large scale in a computerized way by means of data bases and expert systems (Sect. 8.3.6). In particular, a library search is a useful tool to identify pure compounds, conˇrm them and characterize constituents in mixtures. Additionally, unknown new substances may be classiˇed by similarity analysis (Zupan [1986], Hippe [1991], Warr [1993], Hobert [1995]). The library search has its main application in such ˇelds where a large number of components has to be related with large sets of data such as environmental and toxicological analysis (Scott [1995], Pellizarri et al. [1985]). In comparing spectra (and other signal records) with reference data, the uncertainty of measurement of the signal position has to be considered. The comparison of test spectra and reference spectra in form of data ˇles may be regarded from the standpoint of set theory.Inconventional set theory (Cantor [1895]), individual data (elements, here intensity values at given signal positions) follow a membership function m(x) that takes only the values 1 and 0, characterizing an element belonging or not belonging to a given set. Comparing such \hard" data sets, slight shifts of test data

Fig. 2.19. Various sets of analytical data: (A) Hard reference data set, mR(x). (B) Hard test data set, mT (x), which is slightly shifted compared with (A). (C) Fuzzy set of test 2 2 data, mT (x) = exp (x a) =b . (D) Intersection mT R(x) of test data and reference data which is empty in this case. (E) Intersection of fuzzed test data and reference data with a membership value of about 0.8 in this case References 37 (which may be caused by the sample or measurement) result in an empty intersection T \ R of the test and reference data as shown in Fig. 2.19D. Zadeh [1975] extended the classical set theory to the so-called fuzzy set theory, introducing membership functions that can take on any value between 0 and 1. As illustrated by the intersection of the (hard) reference data set (A) and the fuzzed test data set (C), the intersection (E) shows an agreement of about 80%. Details on application of fuzzy set theory in analytical chemistry can be found in Blaffert [1984], Otto and Bandemer [1986a,b] and Otto et al. [1992].

References

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Hartung J, Elpelt B, Klosener K-H (1991) Statistik. Lehr- und Handbuch der angewandten Statistik. R. Oldenbourg Verlag, Munchen, Wien, Kap. XII Harvey CE (1947) A method of semi-quantitative spectrographic analysis. A.R.L., Glendale, CA Hippe Z (1991) Artiˇcial intelligence in chemistry. Structure elucidation and simulation of organic reactions. Elsevier, Amsterdam Hirschfeld T (1980) The hy-phen-ated methods. Anal Chem 52:297A Hobert H (1995) Library search { principles and applications. In: Chemometrics in environmental chemistry { applications. Vol 2, part H (Vol ed J Einax), Springer, Berlin Heidelberg New York, p 1 Horwitz W (1990) IUPAC, Analytical Chemistry Division, Commission on Analytical Nomenclature: Nomenclature for sampling in analytical chemistry (Recommendations 1990), Pure Appl Chem 62:1193 Hulanicki A (1995) IUPAC, Analytical Chemistry Division, Commission on General As- pects of Analytical Chemistry: Absolute methods in analytical chemistry. Pure Appl Chem 67:1905 Inczedy J (1982) Homogeneity of solids: a proposal for quantitative deˇnition. Talanta 29:643 Ingamells CO (1974) New approaches to geochemical analysis and sampling. Talanta 21:141 Ingamells CO (1976) Derivation of the sampling constant equation. Talanta 23:263 Ingamells CO, Switzer P (1973) A proposed sampling constant for use in geochemical analysis. Talanta 20:547 ISO (1993) International Organization for Standardization, Guide to the expression of uncertainty in measurement. Geneva IUPAC Orange Book (1987) Compendium of analytical nomenclature (eds H. Freiser, G.H. Nancollas), Blackwell, Oxford 2nd edn (1st edn 1978) IUPAC Orange Book (1988) Compendium of analytical nomenclature (eds J. Inczedy, J.T. Lengyiel, A.M. Ure, A. Geleneser, A. Hulanicki), Blackwell, Oxford 3rd edn Kateman, G (1990) Sampling Strategies. Wiss Z TH Leuna-Merseburg 32: 54 Kateman G, Buydens L (1993) Quality control in analytical chemistry, 2nd edn. Wiley, New York Keith LH (ed) (1988) Principles of environmental sampling. ACS, American Chemical Society, Washington DC Kellner R, Mermet J-M, Otto M, Widmer HM (eds) (1998) Analytical chemistry. The approved text to the FECS Curriculum Analytical Chemistry. Wiley-VCH, Weinheim, New York, Chichester Kovats E (1958) Gas-chromatographische Charakterisierung organischer Verbindungen. Teil 1: Retentionsindizes aliphatischer Halogenide, Alkohole, Aldehyde und Ketone. Helvet Chim Acta 41:1915 Kraft G (1980) Probennahme an festen Stoffen, in: Analytiker-Taschenbuch, Bd. 1 (Hrsg.: H. Kienitz, R. Bock, W. Fresenius, W. Huber, G. Tolg), Springer, Berlin Heidelberg New York 40 References

Kratochvil B, Wallace D, Taylor JK (1984) Sampling for chemical analysis. Anal Chem 56:114R Kurfurst U (1991) Die direkte Analyse von Feststoffen mit der Graphitrohr-AAS, in: Analytiker-Taschenbuch, Bd. 10 (Hrsg.: H. Gunzler, R. Borsdorf, W. Fresenius, W. Huber, H. Kelker, I. Luderwald, G. Tolg, H. Wisser), Springer, Berlin Heidelberg New York Lucchesi CA (1980) The analytical chemist as problem solver. Internat Lab 11/12:67 Malissa H (1973) Stereometrische Analyse mit Hilfe der Elektronenstrahlmikroanalyse. Fresenius Z Anal Chem 273:449 Markert B (ed) (1994) Environmental sampling for trace analysis. VCH, Weinheim, New York Basel Metzler P, Nickel B (1986) Zeitreihen- und Verlaufsanalyse. Hirzel, Leipzig Moseley HGA (1913) The high-frequency spectra of the elements. Philos Mag 26:1024 Nothbaum N, Scholz RW, May TW (1994) Probenplanung und Datenanalyse bei kon- taminierten Boden . Erich Schmidt, Berlin Otto M, Bandemer H (1986a) Pattern recognition based on fuzzy observations for spectroscopic quality control and chromatographic ˇngerprinting. Anal Chim Acta 184:21 Otto M, Bandemer H (1986b) Calibration with imprecise signals and concentrations based on fuzzy theory. Chemom Intell Lab Syst 1:71 Otto M, Stingeder G, Piplits K, Grasserbauer M, Heinrich M (1992) Comparison of depth proˇles in SIMS by a fuzzy method. Mikrochim Acta 106:163 Parczewski A (1981) The use of empirical mathematical models in the examination of homogeneity of solids. Anal Chim Acta 130:221 Parczewski A, Danzer K (1993) Some limitations of global characterization of inhomogeneity of solids. Polish J Chem 67:961 Parczewski A, Danzer K, Singer R (1986) Investigation of the homogeneity of solids with a linear regression model. Anal Chim Acta 191:461 Pellarrini ED, Hartwell T, Crowder JA (1985) A comparative evaluation of GC/MS data analysis processing. US EPA Project Report PB-85-125664 Prichard E, Green J, Houlgate P, Miller J, Newman E, Phillips G, Rowley A (2001) Analyti- cal measurement terminology { handbook of terms used in quality assurance of analytical measurement. LGC, Teddington, Royal Society of Chemistry, Cambridge Ramendik GI (1990) Elemental analysis without standard reference samples: The general aspect and the realization in SSMS and LMS. Fresenius J Anal Chim 337:772 Scott DR (1995) Empirical pattern recognition/expert system approach for classiˇcation and identiˇcation of toxic organic compounds from low resolution mass spectra. In: Chemometrics in environmental chemistry { applications. Vol 2, part H (Vol ed J Einax), Springer, Berlin Heidelberg New York, p 25 Sherman J (1955) The theoretical derivation of uorescent X-ray intensities from mixtures. Spectrochim Acta 7:283 Singer R, Danzer K (1984) Homogenitatsuntersuchungen von Festkorpern mit Hilfe lin- earer Regressionsmodelle. Z Chem 24:339 References 41

Taggart AF (1945) Handbook of mineral dressing, ores and industrial minerals. Wiley, New York Tolg G (1979) Neue Wege zur analytischen Charakterisierung von Reinststoffen. Fresenius Z Anal Chem 294:1 Tyson JF (1989) Modern analytical chemistry. Anal Proc 26:251 Warr WA (1993) Computer-assisted structure elucidation. Anal Chem 65:1045A; 1087A Wegscheider W (1996) Proper sampling: a precondition for accurate analyses. In: Accredi- tation and quality assurance in analytical chemistry (Ed. by H. Gunzler) Springer, Berlin Heidelberg New York Zadeh LA (1975) Fuzzy sets and their applications to cognitive and decision processes. Academic Press, New York Zupan J (1986) Computer-supported spectroscopic databases. Ellis Horwood, Chichester 3 Signals in Analytical Chemistry

The analytical process is a procedure of gaining information. At ˇrst, samples contain only latent information on the composition and structure, namely by their intrinsic properties (Malissa [1984]; Eckschlager and Danzer [1994]). By interactions between the sample and the measuring system this information is transformed step by step into signals, measured results and useful chemical information.

3.1 Signals and Information

Information is always connected with signals. In general, signals are deˇnite states or processes of material systems (Eckschlager and Danzer [1994]; Danzer [2004]). They can, therefore, be differentiated into static and dynamic signals. Examples of static signals are script, colors, images, ˇgures, and buildings. On the other hand, dynamic signals result from electrical, thermal, optical, acoustic, or chemical interactions. Nowadays, these signals are converted in each case into electric signals, in which form they may be treated and transmitted. Finally, the essential signal characteristics are recorded in a suitable form. The process of signal generation and evaluation is given by the chemical measurement process (CMP, Currie [1985, 1995]) which is shown in its analogy to the information process in Fig. 3.1. Both the CMP and information process in Fig. 3.1 have been simpliˇed in comparison with the representations in Figs. 2.1 and 2.2 because sample preparation has not been considered here and the measurement begins with the measuring sample as information source from which the signals are obtained. From the signal-theoretical point of view, signals must fulˇl the following three functions to gain analytical information:

A syntactic function which describes the relationship between equivalent signals, the formation of signal sequences, and the transformation of signals (knowledge of the genesis of signals). The syntactic function characterizes the structure of signals and signal sequences. A semantic function which describes the meaning and signiˇcance of signals and thus their unambiguous connection with the object to be 44 3 Signals in Analytical Chemistry

Fig. 3.1. Chemical measurement process (above) and information process (below) 1 Mea- surement, 2 Signal validation, 3 Evaluation/calibration, 4 Data evaluation, A Coding, B Selection, C Decoding, data evaluation

characterized (knowledge of coding and decoding, assignment rules, and calibration). The semantic function characterizes the meaning of signals. A pragmatic function which determines the relationship between signals and persons who receive them. The pragmatic function, therefore, characterizes the importance and beneˇt of signals. All three of these functions should be harmonized to obtain the expected information to the full extent and to avoid misinterpretation of signals.

3.2 Analytical Signals

The term \analytical signal" has a different meaning in signal theory and analytical chemistry. In signal theory, analytical signals are of the type z(t)=x(t)+j x^(t) (3.1) where x(t) is the signal function and x^(t)=Hfx(t)g is the Hilbert transform as imaginary part (Wolf [1999]). Here, the term analytical signal is used for all the signals which are produced by analytical methods and used (treated, evaluated and interpreted) in any form in analytical chemistry. Analytical signals can result from test samples, reference samples, or data banks (reference spectra and other recordings). Analytical signals are generated by interactions between species of the analyte, to be precise between certain forms of intrinsic energy of them (see 3.2 Analytical Signals 45

Molecules

Atoms Chemical reactions

Ions (solvatized) Electrochemical processes Electrons

Electron beams Particle–matter interactions (a) Inelastic interactions (energy transfer Ion beams with sample species): SPECTROSCOPY (b) Elastic interactions (energy transfer): Electromagnetic Radiation–matter interactions radiation DIFFRACTION, MICROSCOPY

Heat Thermal interactions

Directed movement Interaction between phases energy (partition, adsorption)

Fig. 3.2. Overview of various forms of interactions taking place between measuring samples and different forms of matter and energy to produce analytical signals

Fig. 2.13), and an external system of matter and energy, respectively. These systems and the resulting interactions are classiˇed in Fig. 3.2. The type of interaction is characterized by the analytical technique applied. In detail, analytical signals can appear in form of: The product of chemical reactions, namely precipitations (their appearance, color, and crystal shape), colors in solution, gases (their appearance, bubbles, and color), and sublimates Changes of colors in ames and other light sources, emission of radiation Differences between physical quantities like temperature, potential, voltage, and absorbance etc. Analytical signals can be classiˇed according to different criteria. Accord- ing to their genesis, analytical signals are frequently differentiated into chemical and physical signals, notwithstanding that chemical signals are quanti- ˇed ultimately by physical quantities (weight, volume, absorbance etc). Analytical signals can also be differentiated according to their time characteristics into dynamic, transient and static signals, and their complexity into single signals, periodic signals, signal sequences and signal functions. Most of the signals with which analysts have to do are manifest signals, i.e. real observable signals from which real measured values can be taken. But it is also possible to evaluate latent signals, i.e. non-real signals which can be obtained mathematically from real signals, in the simplest cases by calculating differences, ratios, and sums of manifest signals. Latent signals are realized in the form of latent variables, e.g. differences of mass numbers in mass spectra as well as isotope pattern of intensities of molecular peaks. A special type of latent variables is represented by principle components, eigenvalues, factors, and other chemometrically obtainable quantities. 46 3 Signals in Analytical Chemistry Hidden signals can be given in the form of non-resolved ˇne structures of spectral bands or signals covered by background and noise. Hidden signals can be detected by improving the experimental technique or by applying chemometric procedures to improve the signal resolution and/or signal-to-noise ratio. In some cases it makes sense to record a signal not only in its original form but also in its integrated form or in the form of its ˇrst, second or higher derivation. In this way, the quantiˇcation of signal intensity can be improved and the position of the signal can be measured more precisely, respectively. The generation of analytical signals is a complex process that takes place in several steps. Methods of instrumental analysis often need ˇve steps, namely (1) the genesis, (2) the appearance, (3) the detection and conversion, (4) the registration, and (5) the presentation of signals; see Fig. 3.3. The genesis of signals is directly connected with the interaction between the entities of the sample (see Fig. 2.13) and the form of matter and energy represented in Fig. 3.2. This interaction produces the signal as a result of a chemical reaction, an electrochemical, physicochemical or physical process, e.g. by a neutralization or precipitation reaction, an electrolytical process, or by interactions between radiation and particles on the one hand and the sample species on the other. The appearance of signals takes place in form of a substance of a given state (gaseous, liquid, precipitate), change of colors by absorption or reection of light, or emitted and absorbed radiation. Spectroscopic signals

METHOD GENESIS APPEARANCE DETECTION CONVERSION REGISTRATION PRESENTATION STORAGE

Emission Interaction Emitted Photo multiplier spectrometry thermal- & radiation tubes, photoplate Spectrum radiation quanta, h⋅ν energy – species Interaction Absorption Absorbed Thermal Pen recorder, Spectral signal, radiation spectrometry radiation conductivity bridge Current display spectrum energy – quanta, h⋅ν species i=f(t) Mass Ionsensitive Ionization Accelerated Spectrum Digitalized spectrometry ions multiplier tubes, data dynodes Nuclear magnetic Interaction Resonance of Radiofrequency Spectrum in time resonance magnetic fields radiation pulses or frequency spectroscopy – nuclei quanta, h⋅ν (FT) domain

Chromatography Interaction Retention at Thermal between phases given time or conductivity, flame- Chromatogram volume ionization, MS Weighing form – Gravimetry Chemical Precipitation ––Result reaction Balance

Titrimetry Chemical Equivalence Volume Pen recorder, Titration curve reaction point Voltage display Electrochemi- Exchange of Potentiometry Electrode potentials E=f(t) Response curve cal reaction charge

Polarography Electrochemi- Exchange of Diffusion current Current, voltage i=f(E) Polarogram cal reaction charge

Fig. 3.3. Illustration of the steps of signal generation for different analytical principles 3.3 Types and Properties of Analytical Signals 47 appear in form of energy quanta h which are characterized qualitatively and quantitatively by wavelengths, frequencies, and intensities. In this form a signal might be visible or not but it cannot be stored, treated or evaluated. Therefore, signals must be brought in an applicable form by a suitable detection procedure. By the detection step the signal is transformed into a form that is electrically measurable. By means of photocells photons and ions can be generated into electrons the number of which is greatly increased in multiplier tubes. Several types of detection are shown in Fig. 3.3. It can be seen that, independently of the genesis (chemical or physical), the manifested signal is always converted into the form of a physical quantity like mass, volume, mostly, however, into an electric quantity (current, voltage). Via the conversion step the signals come into a form where current or voltage can be registered as a continuous function of time or a time-proportional quantity. For the analytical representation the signals have to be transformed from time functions into conventional measuring functions. These are characterized by analytical quantities on abscissa and ordinate axes where the values of them may be relativized in some cases (e.g. MS). Such a transformation of quantities is mostly carried out on the basis of instrument-internal adjustment and calibration. The presentation of signals takes place in the form of records of single signals and of signal functions (displays and prints of spectra, chromatograms, or images). The presentation occurs in the signal domain as schematically shown in Fig. 2.12 on the right side. Because the signal function cannot be reversibly stored in this analogous form, an analogue-to- digital conversion may follow where a sequence of numbers is obtained that is discretely arranged on a time axis. Signals in digital form are resis- tant to disturbances and can be reversibly stored. In particular, there is no limitation in the treatment of digital data. Therefore, some additional procedures can be followed where the signals are chemometrically evaluated and improved, respectively.

3.3 Types and Properties of Analytical Signals

Signals used in analytical chemistry have a deˇnite origin from particular species or given structural relationships between constituents of samples. The relation of the sample domain and the signals domain, i.e. the coding and decoding process as represented in Fig. 2.12, must be as unambiguous as possible. The reliability of signals depends on the information amount (see Chap. 9), particularly on the signal resolving power of the analytical method and, therefore, on the fact how narrow the signals are. The smaller the signal half-width (width of the signal at half the maximum height) is, the more unambiguous are the connections to the belonging species. 48 3 Signals in Analytical Chemistry

This has led to such cases in the history of chemistry that spectroscopic signals have been unidentiˇed till newly discovered elements was found (e.g. rubidium, caesium, indium, helium, rhenium) or new species (highly ionized atoms, e.g. in northern lights [aura borealis], luminous phenomena in cosmic space and sun aura, such as \nebulium", \coronium", \geocoronium", \asterium", which was characterized at ˇrst to be new elements; see Bowen [1927]; Grotrian [1928]; Rabinowitsch [1928]).

There are a lot of forms in which signals are represented in general, and particularly in analytical chemistry. The common form of a signal is bell- shaped (of Gauss-, Lorentz-orVoigt type; see, e.g. Kelly and Horlick [1974]). But there are also various other types of signal forms, e.g. differentiated (ESR, AES), integrated (polarography, NMR), bar diagrams (MS), and dynamic signal functions the form of which may be changing (chromatography). By means of modern instrumental methods, multispecies analysis is carried out where, therefore, signal functions are obtained that contain a certain number of single signals. Signal functions may have a very different character even in case of one and the same analytical method as Fig. 3.4 shows. Figure 3.4 shows (i) a line spectrum (one-dimensional dispersive spectrographic record), (ii) a spectrometric record, (iii) an interferogram obtained by a Fourier transform spectrometer, and (iv, v) two- and three- dimensional double dispersive spectra recorded e.g. by Echelle spectrometers. In principle, all forms may be obtained by OES. In general, signal functions will be obtained in analogous form and consist of a large number of arranged measured points which form a data vector. There are three types of signal functions, which contain:

Exactly one signal for each species or phenomenon provided it is present in the sample (e.g. chromatography) A number of signals per species (e.g. XRF, MS), sometimes up to several thousand per element (OES; see, e.g. Harrison [1939]; De Gregorio and Savastano [1972]) Not only original signals (one or several) but additional combined signals (overtones, coupled oscillations, e.g. NIR) and latent signals in form of relations between original signals (differences, e.g. MS) Occasionally, typical pattern can be observed which can be formed according to special rules like multiplets in ESR-, NMR-, and OES spectroscopy or isotopic ratios in MS (molecular peak pattern). There can also be randomly formed pattern within such spectra, being rich in signals like OES (e.g. the known sodium doublet (Na-D) 589.6 and 589.0 nm, and the magnesium \quintet" 277.67, 277.83, 277.98, 278.14, and 278.30 nm). The identiˇcation of species is always made easier when pattern { whatever type { can be compared instead of a number of signals that are irregularly arranged. Only such signals are used in analytical chemistry, as a rule, which can reliably be related to the species or phenomenon under investigation. To 3.3 Types and Properties of Analytical Signals 49

Fig. 3.4. Different types of signal recordings (schematic) obtained by various types of instruments and registering in OES guarantee the correct recognition and identiˇcation of signals, several procedures are applicable. As mentioned in Sect. 2.4, a reliable evaluation function Q = f 1(z) is the basis of an unambiguous recognition of the analyte. If, nevertheless, uncertainties remain, there are some theoretical and experimental possibilities to guarantee reliably the identity. Such procedures are, e.g. addition of the analyte in a deˇnite form, isotope dilution, and isotope substitution (e.g. deuterization of characteristic groups). Signal functions are inspected in several ways: { By \XX" (expert's experience), or objectively { By comparison with reference lists or diagrams in form of tables or atlases, respectively, and increasingly computerized (mainly in the case of spectra) { By comparison with reference spectra (RS), mainly by automated spectra search (data banks) For library searches, test spectra (TS) have to be available in digitalized form. The demands on the quality of reference spectra are high. They have to 50 3 Signals in Analytical Chemistry be free from systematic deviations and impairing noise and must be unambiguously coded. Both reference and test spectra must be pretreated in the same way concerning background correction, standardization, and encoding. Sometimes it may be advantageous to use test spectra which are fuzzed to a certain degree to compare them successfully with reference spectra (see Sect. 2.4, Fig. 2.19, Blaffert [1984]). Efˇcient techniques which can be applied also for spectra comparisons are subtraction (TS{RS) and cross-correlation of TS and RS (Danzer et al. [1991]). The cross-correlation function (CCF) is calculated according to the relationship +T 1 y y ()=yTS(t) yRS (t + ) = lim yTS(t) yRS (t + )dt (3.2) TS RS T !1 2T T The CCF has a sharp maximum at =0 in case of total agreement of the TS and RS. The correlation coefˇcient then becomes unity. Values less than 1 characterize a certain degree of agreement. Within the signal domain the signal functions can be represented in two sub-domains, namely the original (signal) domain and the image domain which are commonly called time domain and frequency domain. The records of spectra in the time and frequency domain may look like (ii) and (iii) in Fig. 3.4. The transition between the both domains is carried out mathematically by means of the Fourier transformation1. The relations between signal functions in original (time) and Fourier domains (frequency ! =1=t) are represented in Fig. 3.5. Fourier pairs not only exist in time-/frequency domain but also in any other domain combined by a quantity q and the belonging dimension- inverted quantity 1=q. Signals are characterized by typical parameters. In Fig. 3.6 the fundamental signal parameters of a common bell-shaped signal are given. Signals are determined by the following parameters:

Position zA: the quantity which corresponds to the energy or motion, respectively, of species of the analyte A. As detailed in Sect. 3.5, the signal position is not a totally ˇxed parameter but is variable to a certain degree.

Intensity yA: the quantity which corresponds to the amount of the analyte

A (species or phenomenon). Both maximum intensity, yAmax , and integral

intensity, yAint , are used depending on the analytical procedure. In the

case of maximum intensity, frequently the net intensity yAnet

yAnet = yAmax y0 (3.3)

1 There are other transformations, too, e.g. Laplace transformation which is frequently used in technical systems. 3.3 Types and Properties of Analytical Signals 51

Signal function in the time domain Signal function in the frequency domain +∞ F +∞ 1 ω {y(t)} − ω y(t) = Y ( jω) e j tdω Y ( jω) = y(t) e j tdt 2π −∞ −∞ F–1{Y(jω)}

Time average y(t) y(t +τ ) = 2 1 Y( jω) +T S (ω) = lim Ψ τ = 1 +τ YY π →∞ yy ( ) lim y(t) y(t )dt 2 T 2T T →∞ 2T −T

+∞ F{y(t)} +∞ Ψ τ = ω jω t ω ω = 1 Ψ τ − jω t τ yy ( ) SYY ( ) e d SYY ( ) yy ( ) e d −∞ 2π −∞ F–1{Y(jω)} Autocorrelation function Power spectrum (Spectral power density)

Fig. 3.5. Connection between characteristic signal functions in time and frequency domain

Fig. 3.6. Essential signal parameters: zA signal position, yA signal (gross) intensity, y0 signal background, fwhm(z) full width at half maximum (signal half width), fwb(z) full width at background 52 3 Signals in Analytical Chemistry which characterizes the number of analyte species in the measuring sample, is applied. The integral intensity corresponds to the area under the signal and is estimated by

fwb (z)

yAint = y (z)dz (3.4)

In some cases, the integral intensity is recorded additionally to the original signal function. Shape, symmetry, and width: these parameters can be inuenced both by properties of the analyte species and their interactions, e.g. with neighboring species, and by disturbing effects of the measuring system (e.g. tailing effects in chromatography and peak widening in general). Peak shape can be considered as a convolution of the \true" signal function and the measuring function (see Fig. 2.15). Fine structure and splitting: under given conditions of the measuring technique (e.g. state of the sample, high-resolution instruments, magnetic ˇelds) signals may produce a ˇne structure or split up to multiplet pattern. Splitting follows given rules. Background and noise: sometimes signals are affected by a background (zero signal) y0 which can be corrected experimentally or by calculation. On the other hand, noise is an inherent component of signals, and may appear weak or strong. Noise has effects in both z- and y -directions and causes random variations of the analytical quantities z and y . Noise in the z-direction is mostly and rightly disregarded but can become signiˇcant in special cases, e.g. in remote sensing where laser pulses may be blurred in both z- and y -direction (Kozlov [2004]).

A signal function y (z), which is mostly treated as a time-dependent function, y (t), can be regarded as consisting of the original \true" signal function ytrue(t) which is superposed by a noise function n(t); see Fig. 2.15:

y (t)=ytrue(t)+n(t) (3.5) Noise is generally added to the signal function; the types of noise are schematically shown in Fig. 3.7. In analytical instruments, all types of noise can be produced simultaneously { white and icker noise as well as interference. The following procedures can be applied to improve the signal-to-noise ratio S=N: signal averaging (accumulation), digital ˇltering (Williams [1986]), autocorrelation, and smoothing of the signal function. The applicability of these techniques depends on the type and form of the power spectra of both signal and noise. The only method that can be universally applied, is signal averaging.p If the signal function is measured n times, the S=N ratio increases by n. 3.4 Dimensionality of Analytical Signals and Information 53

Fig. 3.7. Power spectra of the most important types of noise (schematic representation in the frequency domain)

3.4 Dimensionality of Analytical Signals and Information

The dimensionality of a functional relationship will be deˇned here (ax- iomatically) by the number of dependent and independent variables in such a function. Therefore, functions of the type a = f (b) are two-dimensional,of the type a = f (b1,b2) three-dimensional, and of the type a = f (b1;b2;:::bn) are (n +1)-dimensional. The representation of various realizations of only one variable (either y or z)isone-dimensional (Danzer et al. [2002]). Dimensionality in analytical chemistry comprises different types of dimensions displayed by different sorts of variables, which can be discrete or continuous:

Chemical dimensions (dimensions in the sample domain): type (Q), number (n) and amount (x) of analytes, i.e. distinct chemical species; see Fig. 2.12 (left) Measuring dimensions (signal-theoretical dimensions in the signal domain): signal position z (given by the signal's energy- or time characteristics) and signal intensity y ; see Fig. 2.12 (right) Physical dimensions: time (t) and space, characterized by the variables latitude (lx ), longitude (ly ), and altitude (lz ) Statistical dimensions: number of variables (manifest or latent) taken into account in evaluation. Statistical dimensions deˇne the type of data handling and evaluation, e.g. univariate, bivariate, multivariate As a result of analytical measurements, signals are obtained and, in the case of instrumental measurements, signals functions, y = f (z). The record of the signal intensity as a function of the signal position, Fig. 3.8, represents a two-dimensional signal function which can be back-transformed into two- dimensional analytical information, x = f (Q). The abscissa in Fig. 3.8 may represent an energy-related scale, e.g., wavelength-, frequency-, or mass/charge coordinates of spectrometers or re- 54 3 Signals in Analytical Chemistry

Fig. 3.8. Two-dimensional analytical information in form of a signal function tention time coordinates of chromatograms (Danzer and Currie [1998]). The evaluation of such signal functions is carried out in both dimensions as a rule. An evaluation of only one of the two quantities concerns:

(a) The intensity yA of a given signal in position zA which is known to be characteristic for a certain species A; see Fig. 3.9 (left-hand side)

(b) The appearance of certain signals zi that indicate the presence of the corresponding species in the sample (Fig. 3.10)

In this case, the z-axis is inspected as to whether a signal (and therefore the belonging species) is found (+) or not (). Therefore, one-dimensional analytical evaluation of signals corresponds to the quantitative analysis of one given analyte by typical single species methods like AAS, as shown in Fig. 3.9 (left side) on the one hand, and qualitative multielement analysis, frequently carried out in form of inspection analysis, by typical multielement methods like OES. The basis of testing the presence of a certain species Q Currie involves the exceeding of the critical value, yQ >yQcrit ( [1995]). Quasi-multidimensional information as shown in Fig. 3.9 (right side) can be obtained in different ways, mostly by sequential measurements, for example:

By changing detecting channels, thus, measuring one analyte after another (e.g., spectrophotometry) By changing excitation source (e.g., hollow cathode lamps in AAS) By measurement the same element, which is temporally separated in different species (e.g. GC-AAS)

The most frequent case in analytical chemistry is the evaluation of two- dimensional signal functions in the form of spectra, chromatograms, ther- mograms, current-voltage curves, etc. (Fig. 3.8). 3.4 Dimensionality of Analytical Signals and Information 55

Fig. 3.9. One-dimensional information (in y -direction) of a single signal at a ˇxed z-value (left-hand side) and quasi-multidimensional information on several analytes A;B;:::;N as a sequence of one-dimensional information (right-hand side)

Fig. 3.10. One-dimensional information in z-direction (signal inspection): qualitative evaluation, signal identiˇcation

Additional dimensions in signal functions can be given by:

Successive separation steps, e.g. in two-dimensional chromatography (two-dimensional thin-layer chromatography) that result in three- dimensional signal functions y = f (z1;z2), as schematically shown in Fig. 3.4(v). Two-dimensional excitation experiments (two wavelengths excitation in uorescence spectroscopy or two frequency experiments in 2D-NMR) also generate three-dimensional signal functions. Time-dependent analytical measurements, which give three-dimensional information of the type y = f (z;t) as shown schematically in Fig. 3.11a. The same characteristic holds for distribution analysis in one spatial direction, i.e., line scans, y = f (z;lx ). Such signal functions are frequently represented in form of multiple diagrams as shown in Fig. 3.11b. 56 3 Signals in Analytical Chemistry

Fig. 3.11. Temporally (t) or spatially (l) dependent analytical measurements, e.g., a line scan of electron microprobe analysis represented in a three-dimensional (a) and quasi- three-dimensional way (b)

Distribution-analytical investigations in which spatial directions of the sample represent additional variables such as:

{ Line scans of microprobes y = f (z;lx ) for several analytes giving images like that represented in Fig. 3.11

{ Surface scans of microprobes y = f (z;lx ;ly ) for a given analyte, see Fig. 3.12a,b

{ Component images z = f (lx ;ly ) for one or two analytes by black and white representations like Fig. 3.12c, for more components by color images { So-called \3D" images of SIMS, which in fact represent four- or ˇve- dimensional analytical information depending on the actual function y = f (lx ;ly ), y = f (lx ;ly ;lz ), or y = f (z;lx ;ly ;lz ), respectively Coupled (hyphenated) analytical techniques of a separation method and a method of analytical determination, e.g. GC-MS, see Fig. 3.13, by which higher-dimensional signal functions are obtained. Strictly speaking, the dimension is 2(n + 1) in the case that n mass spectra are recorded.

Alternative characterization of the dimensionality. The SIMS example demonstrates that the dimensionality of analytical information and of signal functions occasionally follow other principles than those given above, where the dimensionality of a functional relationship is determined by the number of dependent and independent variables in such a function. 3.4 Dimensionality of Analytical Signals and Information 57

Fig. 3.12. Different types of surface scans: a two- dimensional isoline representation of the Mn distribution on a steel surface measured by M-OES; b the same Mn distribution in a three-dimensional representation (Danzer [1995]); c EPMA scan of Si in a binary eutectic Al-Si alloy

Fig. 3.13. Higher-dimensional system of signal functions in case of coupled techniques, e.g., GC-MS 58 3 Signals in Analytical Chemistry

Fig. 3.14. Line with a fractal dimension of about 1.8

Table 3.1. Selection of signal functions in analytical chemistry and their dimensionality

Signal function Information Dimension

zy;lx ;ly ;lz ;t Qualitative analysis (y , lx , ly , lz , t=const) for several 1 analytes, e.g., by a sequence of spot tests

yz;lx ;ly ;lz ;t Quantitative average analysis (z, lx , ly , lz , t=const) for 1 one given analyte, e.g., by gravimetry or AAS

y = f (z) lx ;ly ;lz ;t Quantitative average analysis (lx , ly , lz , t=const) for sev- 2 eral analytes, e.g., by ICP-OES

y = f (lx )z;ly ;lz ;t Line analysis: determination of the variation (distribu- 2 tion) of the amount of one analyte (given z) along a line lx across the sample, e.g., by EPMA

y = f (t)z; lx ;ly ;lz Quantitative process analysis: recording the amount of 2 one given analyte z in dependence of time, e.g., by ow techniques

y = f (z;lx )ly ;lz ;t Line analysis of several analytes: determination of the 3 variation (distribution) of the amounts along lx , see Fig. 3.11, e.g., by EPMA

z = f (lx ;ly )y;lz ;t Surface analysis: qualitative representation of the pres- 3 ence (distribution) of several analytes across a surface area, see Fig. 3.12c obtained by EPMA

y = f (lx ;ly )z;lz ;t Surface analysis: quantitative representation of the vari- 3 ation (distribution) of the amount of a given analyte z across a surface area, see Fig. 3.12b obtained by micro spark OES

y = f (lx ;ly ;lz )z;t Volume distribution of one given analyte: SIMS data 4 (representation mostly by categorial y , e.g., color- coded) . . . .

y = f (z;lx ;ly ;lz ;t) Quantitative determination of several analytes in de- 6 pendence of space and time: spatial-resolved dynamic quantitative multicomponent analysis, e.g., by a series of 3D-SIMS images for several elements and at different times 3.4 Dimensionality of Analytical Signals and Information 59 In contrast with this, in signal theory the dimensionality of signal functions is frequently determined only by the number of independent variables. The dependent quantities are ignored. Therefore, a signal curve representing the dependence of the signal intensity from a given independent variable (mostly time) is characterized as being one-dimensional. Also, in analytical practice such a characterization of analytical functions is sometimes used, especially in relation to analytical images. So, spectra are sometimes considered as one-dimensional images and analytical element distribution like those given in Fig. 3.12b,c as two-dimensional element images. A different point of view is also adopted by Booksh and Kowalski [1994], who start from an order of instruments or methods in such a way that a classiˇcation is given according to the type of data which is generated. An instrument (or method) is called a zero-order instrument if it generates a single datum per sample. This terminology comes from mathematics where a single number is considered a zero-order tensor. Examples for zero- order instruments are ion sensitive electrodes (ISE) and single-wavelength photometers. First-order instruments are represented by common spectrometers, chromatographs and similar equipment. They usually produce data vectors (e.g., spectral intensities at multiple wavelengths). Second-order instruments yield a data matrix per sample as is the case in coupled (hyphenated) techniques like GC-MS, MS-MS, HPLC-MS, etc. Similar to the way of looking at dimensions in signal theory, the tensorial theory of instruments neglects one of the dimensions. But whereas in signal theory the ordinate values (e.g., the intensity y ) are ignored, in tensorial theory the abscissa values z (e.g., wavelengths or retention times) are disregarded. Notwith- standing, the classiˇcation according to the order of instruments and methods, respectively, represents a suitable basis for theoretical fundamentals of analytical chemistry.

From the most general point of view, the theory of fractals (Mandelbrot [1977]), one-, two-, three-, m-dimensional ˇgures are only borderline cases. Only a straight line is strictly one-dimensional, an even area strictly two-dimensional, and so on. Curves such as in Fig. 3.11 may have a fractal dimension of about 1.1 to 1.3 according to the principles of fractals; areas such as in Fig. 3.12b may have a fractal dimension of about 2.2 to 2.4 and the ˇgure given in Fig. 3.14 drawn by one line may have a dimension of about 1.9 (Mandelbrot [1977]). Fractal dimensions in analytical chemistry may be of importance in materials characterization and problems of sample homogeneity (Danzer and Kuchler [1977]).

Analytical functions of different dimensionality have been listed in some detail by Danzer et al. [1987]. The most important signal functions of practical relevance are given in Table 3.1.

Dimensionality of analytical data. Analytical data are present either in the form of measured values yi or analytical results xi. Multivariate data, i.e., results of m variables (e.g., analyte concentrations) measured at n different samples, are mostly represented in the form of data sets and data matrices: 60 3 Signals in Analytical Chemistry ⎛ ⎞ x11 x12 x1m ⎜ ⎟ ⎜ x21 x22 x2m ⎟ X = ⎝ . . . ⎠ (3.6) . . . xn1 xn2 xnm where m is the number of components (distinct analytes) determined in n objects (samples). The matrix at Eq. (3.6) has the dimension n m, and the related data set is called m-dimensional. It is the aim of multivariate data analysis to reduce the dimensions of data sets (matrices) by exclusion of redundant variables (components which are correlated) up to such a number that can be displayed two- or three-dimensionally. Details of how to achieve reduction of dimensionality in data sets will be given in Sect. 8.3.1 and are available in the chemometric literature (e.g., Sharaf et al. [1986]; Massart et al. [1988]; Frank and Todeschini [1994]; Danzer et al. [2001]).

3.5 Mathematical Model of Signal Generation

In an ideal case, the signal yA = f (zA), as shown in Fig. 3.6, is determined only by the analyte A (or the phenomenon of interest), namely both the position, zA = f (A), and intensity, yA = f (xA). But in real samples, matrix constituents are present which can principally interfere with the analyte signal. In structure analysis the same holds for the neighboring relationships (the \environment" of the species A of interest). Therefore, signal parameters are additionally inuenced by the matrix (or the \neighborhood", respectively), namely the species B;C;:::;N, and follow then the complex relationships zA = f (A; B;C;:::;N), yA = f (xA; xB;xC ;:::;xN ). Additionally, inuencing factors a;b;:::;m, background, y0, and noise (random deviations eA) may become relevant and have to be considered. The signal position is given by a characteristic value zA0 which is determining for the species A according to evaluation rules for identiˇcation and qualitative analysis; see Sect. 2.4, Eq. (2.18a{c). Additional changes in position such as: { Chemical shifts z { Fine structures such as rotation structure of vibration bands { Multiplet splitting can arise from neighboring effects, interactions, and couplings, in accor- dance with natural laws and rules (multiplet structure). Inuencing factors such as temperature, pressure, solvent, etc., can also alter the signal position which holds in general:

zA = f (A; B;C;:::;N; a;b;:::;m) (3.7) The real signal position and structure then is given by 3.5 Mathematical Model of Signal Generation 61 ⎧ ⎫ ⎨ N m ⎬ zA = M ⎩zA0 + ızAi + zAj ⎭ (3.8) i=B j=a where M is the multiplet structure according to relevant rules. In the case of even-numbered multiplets, zA0 can become latent and is then represented by the mean of the multiplet. The sum of ızAQ comprises the resulting chemical shift, that of zAj , the additional shift by inuencing factors. The signal intensity is inuenced by the entire matrix (Q = A;B;C;:::;N) with preference of A. In ideal case, yA would be only caused by xA:

yA = f (xA)=SAA xA + eA (3.9) where the factor SAA represents the sensitivity SAA = @yA=@xA of the determination of A; see Sect. 7.2. But in reality, the accompanying species i = B;C;:::;N can contribute to the signal intensity to variable degree: N yA = f (xA; xB;xC ;:::;xN )=SAA xA + SAixi + eA (3.10) i=B

The factors SAi represent the cross sensitivities (partial sensitivities according to Kaiser [1972]) characterizing how the signal y (zA) is inuenced by the species i:

@yA SAi = (3.11) @xi

The deviation eA represents the unavoidable experimental error of the determination of A. Only in exceptional cases is it possible to estimate all or part of the inuences of interfering species because their amount is mostly unknown. In such cases, the sum of SAixi is considered an additional term of deviation ei caused by the interferents:

yA = f (xA; xB;xC ;:::;xN )=SAA xA + ei + eA = SAA xA + eAi (3.12) where eAi = ei + eA is the summarized deviation term of the deviations of interferences ei and the experimental deviations. In addition, the signal intensity is inuenced by effects of various factors a;b;:::;mof the operating conditions, e.g., temperature, pressure, pH value, and instrumental adjustments. Taking these factors into account the model becomes

yA = f (xA; xB;xC ;:::;xN ; a;b;:::;m) N m = SAA xA + SAixi + IAj xj + eA (3.13) i=B j=a

The quantity IAj represent the speciˇc strength of the inuence factor j on the signal yA, IAj = @yA=@xj . Their products with the given inuence 62 3 Signals in Analytical Chemistry values xj yield the actual factor effects. Analogous to the interferences i, the inuences of the factors j are also hard to quantify. It can be done by means of a multifactorial design on the basis of a model like Eq. (3.13); see Sect. 5.1. But usually the sum of inuences is also considered as a further additive term of deviations ej :

yA = f (xA; xB;xC ;:::;xN ; a;b;:::;m)

= SAA xA + ei + ej + eA = SAA xA + eAij (3.14)

The signiˇcance of inuences, namely both the interferences and several factors can be studied in two ways:

(i) By hypothesis testing where the null hypotheses H0: ei = 0, and H0: ej = 0, differently formulated H0: eAi = eA,H0: eAj = eA, and H0: eAij = eA are veriˇed by Fisher's F -test (see Sect. 4.3.3). In this way, all the factors can only be summarily tested. (ii) By experimental design each inuence variable can be examined separately but in interaction to the other variables. This can be done by using a multifactorial design according to a linear model that contains all the factors of interest (see Sect. 5.1). Mostly models of the type

yA = f (xB;xC ;:::;xN ; a;b;:::;m) (3.15)

are applied, i.e., the study is carried out at a constant analyte concentration. The values of the inuence factors then are varied between sensible levels. The differences of the levels should be, on the one hand, large enough to recognize the effects and, on the other hand, not too large to avoid concealing of (mostly nonlinear) effects.

Taking into account that an experimental blank yA0 has to be considered, Eqs. (3.12){(3.14) can be summarized as follows:

Gross Blank Analyte Signal contributions by Random signal signal other species inuence factors deviations N m yA = yA0 + SAAxA + SAixi + IAixj +eA0 (3.16a) i=B j=a yA = yA0 + SAAxA +ei +ej +eA0 (3.16b) yA = yA0 + SAAxA +eAij (3.16c)

Taking into account that Eqs. (3.16) may also contain some unknown inuences uk which reveal themselves in interlaboratory comparisons but may normally be hidden in the term eA0 , the fundamental signal model is given by References 63

N m yA = yA0 + SAA xA + SAixi + IAj xj + uk + eA (3.17) i=B j=a k

The terms of the signal model at Eq. (3.17) are of particular interest for both theoretical and practical aspects of analytical chemistry, viz. in detail:

y (i) A0 Blank see Sect. 6.2 Detection limit Sect. 7.5 (ii) SAA Sensitivity Sect. 7.2

N (iii) ∑S x Strength of interferences Ai i S Specificity Sect. 7.3 i=B Ai Cross sensitivity

m Robustness Sect. 7.4 (iv) ∑ I x Strength of influences Aj j I Sect. 5.1 j=a Aj Specific influence

Ruggedness Sect. 7.4 (v) ∑uk Unknown influences k

(vi) eA Random deviation Precision Sect. 7.1.1

In the following chapters, these analytical quantities and parameters will be considered in more detail in the given relationships.

References

Blaffert T (1984) Computer-assisted multicomponent spectral analysis with fuzzy data sets. Anal Chim Acta 161:135 Booksh KS, Kowalski BR (1994) Theory of analytical chemistry. Anal Chem 66:782A Bowen IS (1927) The origin of the nebulium spectrum. Nature 1927:473 Currie LA (1985) The limitations of models and measurements as revealed through chemometric intercomparisons. J Res NBS 90:409 Currie LA (1995), IUPAC, Analytical Chemistry Division, Commission on General Aspects of Analytical Chemistry. Nomenclature in evaluation of analytical methods including detection and quantiˇcation capabilities. Pure Appl Chem 67:1699 Danzer K (1995) Sampling of inhomogeneous materials. Statistical models and their practical relevance. Chem Anal [Warsaw] 40:429 Danzer K (2004) A closer look at analytical signals. Anal Bioanal Chem 380:376 Danzer K, Currie LA (1998), IUPAC, Analytical Chemistry Division, Commission on Gen- eral Aspects of Analytical Chemistry. Guidelines for calibration in analytical chemistry. Part 1. Fundamentals and single component calibration (IUPAC Recommendations 1998) Pure Appl Chem 70:993 64 References

Danzer K, Kuchler L (1977) Homogenitatskriterien fur festkorperanalytische Untersuchun- gen.Talanta 24:561 Danzer K, Eckschlager K, Wienke D (1987) Informationstheorie in der Analytik. I. Grund- lagen und Anwendung auf Einkomponentenanalysen. Fresenius Z Anal Chem 327:312 Danzer K, Wienke D, Wagner M (1991) Application of chemometric principles for evaluation of emission spectrographic data. Acta Chim Hungar 128:623 Danzer K, Hobert H, Fischbacher C, Jagemann K-U (2001) Chemometrik { Grundlagen und Anwendungen. Springer, Berlin, Heidelberg, New York Danzer K, van Staden JF, Burns DT (2002), IUPAC, Analytical Chemistry Division, Com- mission on General Aspects of Analytical Chemistry. Concepts and applications of the term \dimensionality" in analytical chemistry (IUPAC Technical Report) Pure Appl Chem 74:1449 De Gregorio P, Savastano G (1972) Spektrum des Eisens von 2206 bis 4656ª mit Analy- senlinien. Specola Vaticana Eckschlager K, Danzer K (1994) Information theory in analytical chemistry. Wiley, New York, Chichester, Bribane Frank IE, Todeschini R (1994) The data analysis handbook. Elsevier, Amsterdam Grotrian W (1928) Uber den Ursprung der Nebellinien. Naturwiss 16:177 Harrison GR (1939) M. I. T. wave-length tables with intensities in arc, spark or discharge tube of more than 100,000 spectrum lines. Wiley, New York Kaiser H (1972) Zur Deˇnition von Selektivitat, Speziˇtat und Empˇndlichkeit von Ana- lysenverfahren. Fresenius Z Anal Chem 260:252 Kelly PC, Horlick G (1974) Bayesian approach to resolution with comparisons to conventional resolution techniques. Anal Chem 46:2130 Kozlov VV (2004) Quantum regulations for trafˇc on intercontinental Telecom highways. Humboldt Kosmos 84:31 Malissa H (1984) Analytical chemistry today and tomorrow. Fresenius Z Anal Chem 319:357 Mandelbrot BB (1977) Fractals { form, chance, and determination. Freeman, San Fran- cisco, CA Massart DL, Vandeginste BGM, Deming SN, Michotte Y, Kaufman L (1988) Chemometrics { a textbook. Elsevier, Amsterdam Rabinowitsch E (1928) Physikalische Methoden im chemischen Laboratorium. IV. Bedeu- tung der Spektroskopie fur die die chemische Forschung. Z Angew Chem 16:555 Sharaf MA, Illman DL, Kowalski BR (1986) Chemometrics. Wiley, New York Williams CS (1986) Designing digital ˇlters. Prentice Hall, New York Wolf D (1999) Signaltheorie. Springer, Berlin Heidelberg New York 4 Statistical Evaluation of Analytical Results

Both qualitative observations and quantitative measurements cannot be reproduced with absolute reliability. By reason of inevitable deviations, measured results vary within certain intervals and observations, mostly in form of decision tests, may fail. The reliability of analytical tests depends on the sample or the process to be controlled and the amount of the analyte, as well as on the analytical method applied and on the economical expenditure available.

4.1 Reliability of Analytical Observations and Measurements

Analytical measurements are fundamentally subject of uncertainty where various types of deviations (errors) can appear and these may be inuenced to varying degree. Even when instrument readings are sufˇciently accurate, repeated measurements of a sample lead, in general, to measured results which deviate by varying amounts from each other and from the true value of the sample. Before the different types of deviations are characterized in detail, some essential terms have to be considered. In analytical chemistry, the term error (used in the sense of deviation) is deˇned by the difference between the test result (xtest) and the true value (xtrue, i.e., the accepted reference value, see ISO 3534-1 [1993]; Fleming et al. [1997]). The term may be related both to measured value y and analytical value x which correspond to each other according to the sensitivity factor b of an analytical procedure. According to their character and magnitude, the following types of deviations can be distinguished. Random deviations (errors) of repeated measurements manifest themselves as a distribution of the results around the mean of the sample where the variation is randomly distributed to higher and lower values. The expected mean of all the deviations within a measuring series is zero. Random deviations characterize the reliability of measurements and therefore their precision. They are estimated from the results of replicates. If relevant, it is distinguished in repeatability and reproducibility (see Sect. 7.1) 66 4 Statistical Evaluation of Analytical Results Systematic deviations (errors) displace the individual results of measurement one-sided to higher or lower values, thus leading to incorrect results. In contrast to random deviations, it is possible to avoid or eliminate systematic errors if their causes become known. The existence and magnitude of systematic deviations are characterized by the bias. The bias of a measured result is deˇned as a consistent difference between the measured value ytest and the true value ytrue:

bias(y )=ytest ytrue (4.1a) The absence of a bias characterizes the accuracy of a measured result. The same holds for the analytical value:

bias(x)=xtest xtrue (4.1b) Outliers are random errors in principle. However, they have to be eliminated because of their disproportionate deviation, so that the mean will not be misrepresented. Gross errors are generated by human mistakes or by instrumental or computational error sources. Depending on whether they are short- or long- term effects, they may have systematic or random character. Frequently, it is easy to perceive and to correct for them. They will not play any role in the following discussion. The relation between systematic and random deviations as well as the character of outliers is shown in Fig. 4.1. The scattering of the measured values is manifested by the range of random deviations (conˇdence interval or uncertainty interval, respectively). Measurement errors outside this range are described as outliers. Systematic deviations are characterized by the relation of the true value and the mean y of the measurements, and, in general, can only be recognized if they are situated beyond the range of random variables on one side. Conventionally, a measured result is said to be correct if the true value is situated within the conˇdence interval of the observed mean (1, case 1 in Fig. 4.1). If the true value is located outside of the range of random deviations (2, case 2 in Fig. 4.1) then the result is incorrect.

Fig. 4.1. Schematic representation of systematic and random deviations 4.1 Reliability of Analytical Observations and Measurements 67 It is not always possible to tell strictly the difference between random and systematic deviations, especially as the latter are deˇned by random errors. The total deviation of an analytical measurement, frequently called the \total analytical error", is, according to the law of error propagation, composed of deviations resulting from the measurement as well as from other steps of the analytical process (see Chap. 2). These uncertainties include both random and systematic deviations, as a rule.

4.1.1 Systematic Deviations Systematic deviations, in addition to random deviations, may be produced at all steps of the analytical process, to be precise, for example during { Sampling by incorrect (non-representative) treatment of individual sample fractions. { Sample preparation by incomplete dissolution, separation or enrichment operations. { Measurement caused by concurrent reactions or incomplete reaction processes in the case of chemical principles, and by instrumental deviations and wrong adjustment in the case of physical methods. A frequently encountered reason for the occurrence of systematic deviations is erroneous calibration due to unsuitable calibration standards, matrix effects, or insufˇcient methodical or theoretical foundation. { Even data evaluation, often thought free of errors to a large extent, can generate systematic deviations by reason of incorrect or incomplete algorithms. According to their inuence on the measurand, one can distinguish (see Fig. 4.2): (i) Additive deviations altering the measured values by a constant value ˛. Instead of the true value xtrue, the falsiˇed value

xtest = xtrue + ˛ (4.2)

is measured. A reason may be, for example, an unrecognized blank. (ii) Multiplicative deviations, proportional to the measured value and changing the slope of the calibration curve and thus sensitivity, because, instead of the true value y the misrepresented value

xtest = ˇ xtrue (4.3)

is measured. They are often generated by erroneous calibration factors. (iii) Nonlinear, response-depending errors causing an incorrect value

xtest = (xtrue) (4.4) 68 4 Statistical Evaluation of Analytical Results

Fig. 4.2. Effects of systematic errors on test values according to Eqs. (4.2){ (4.4); the line of 45ı (broken) corresponds to ideal recovery (line of true measurements, xtest = xtrue)

to be measured instead of the true value. As a result, the calibration linear relationship between measured value and analytical value is lost. In OES, for example, self-absorption of resonance signals generates such an effect. Several of the types of systematic deviations mentioned can frequently occur in combination with each other: xtest = ˛ + ˇ (xtrue) (4.5)

In analytical practice, they are best recognized by the determination of xtest as a function of the true value xtrue, and thus, by analysis of certiˇed reference materials (CRMs). If such standards are not available the use of an independent analytical method or a balancing study may provide information on systematic errors (Doerffel et al. [1994]; Kaiser [1971]). In simple cases, it may be possible, to estimate the parameters ˛, ˇ, and , in Eq. (4.5) by eliminating the unknown true value through appropriate variation of the weight of the test portions or standard additions to the test sample. But in the framework of quality assurance, the use of reference materials is indispensable for validation of analytical methods. Commonly the absence of systematic errors is tested by recovery studies. The estimation of recovery by analyzing only one standard, as frequently be done, can give misleading results as can be seen from Fig. 4.2. In case of combined and nonlinear systematic deviations there can occur points of accidental congruence that can be hit. Therefore, if at all possible, there should be estimated the entire recovery function. However, at least two different standards should be analyzed. 4.1 Reliability of Analytical Observations and Measurements 69 4.1.2 Random Variations

Repeated measurements of the same measurand on a series of identical measuring samples result in random variations (random errors), even under carefully controlled constant experimental conditions. These should include the same operator, same apparatus, same laboratory, and short interval of the time between measurements. Conditions such as these are called repeatability conditions (Prichard et al. [2001]). The random variations are caused by measurement-related technical facts (e.g., noise of radiation and voltage sources), sample properties (e.g., inhomogeneities), as well as chemical or physical procedure-speciˇc effects. By careful proceeding of measurements random variations can be minimized, but fundamentally not eliminated. The appearance of random errors follow a natural law (often called the \Gauss law"). Therefore, random variations may be characterized by mathematical statistics, namely, by the laws of probability and error propagation. Classifying varying measured values by their magnitude does not, as a rule, result in a uniform distribution over the whole variation range, but gives rise to a frequency distribution around the mean value, as shown, e.g., by the bar graph in Fig. 4.3a. Increasing the number of repeated measurements to inˇnity, while decreasing more and more the width of classes (bars), normally leads to a symmetrical bell-shaped distribution of the measured values, which is called Gaussian or normal distribution. The frequency density of the measured values p(y ) shown in Fig. 4.3b is given by the relation 1 (y )2 p y p (y )= exp 2 (4.6a) y 2 2y with the parameters y being the maximum (mean value) and y being half the distance of the inection points of the curve (standard deviation). The same characteristics hold for the analytical values 1 (x )2 p x p (x)= exp 2 (4.6b) x 2 2x where their parameters x and x are related to that of y according to the linear calibration function (see Chap. 6). Using standardized values with z =(x x )=x or z =(y y )=y leads to 1 1 p (z)=p exp z2 (4.6c) 2 2 with a mean of zero and a standard deviation of 1, see Fig. 4.3d. In analytical practice, not the population but random samples are studied. 70 4 Statistical Evaluation of Analytical Results

Fig. 4.3. Schematic frequency distribution of measured values y (a), Gaussian normal distributions of measured values y (b) as well as of analytical values x(c), and standard normal distribution (d)

The term \sample" is used here in a statistical way and not in chemical sense as mentioned above (Sect. 2.1). A sample is a part of the population taken from it by random sampling which has to be representative. Examples from the daily life are predictions by TED (TEle Dialogue) for election results or popularity of persons or groups of people. In a technical process the population may be given by the total daily production. For the reason of quality assurance several a day test samples are taken from which conclusions on the total quality are derived. To distinguish between population and samples from the statistical point of view, different symbols are used. Greek symbols (, ) characterize the population and normal letters (e.g., x, s) the sample. The distributions of samples differ from the normal distribution and are characterized instead of this by the so-called t-distribution (Student [1908]). Figure 4.4 shows the relation between the normal distribution of a given population and some possible sample distributions. 4.1 Reliability of Analytical Observations and Measurements 71

p(y)

Fig. 4.4. Distribution of the population (parents distribution) and the dis- y y y yy 1 2 3 4 tributions of four samples µ taken from this population

The parameters of the normal distribution (the population parameters) are calculated as follows, namely the mean N 1 = y (4.7) y N i i=1 and the variance N 2 (yi y ) 2 = (4.8) y N i=1 where N is the total number of objects of the population. On the other hand, the sample parameters which represent the estimates of and 2, respectively, are calculated by n 1 y = y (4.9) n i i=1 and n (y y )2 s2 = i (4.10) y n 1 i=1 where n is the number of objects (sub-samples) of the respective sampling procedure. The variances 2 and s2, respectively, are additive quantities. Therefore, the variance of a mean y is given by 2 sy s2 = (4.11) y n From the variance some measures of dispersion are derived. The most commonly used are 72 4 Statistical Evaluation of Analytical Results

Standard deviation as the square root of the variance n 2 (yi y ) s = i=1 (4.12) y n 1

The standard deviation can also be obtained from repetition measurements of different samples with various contents: nA m 2 (yij y i) i=1 j=1 s = (4.13) y n m

where m is the number of samples (series), nA the number of repetitions for each of the m samples and with it n = m nA the total number of determinations (yij single measured values, y i sample means (means of series), the number of degrees of freedom is df = n m in this case). For this proceeding, the standard deviation of the samples should not differ signiˇcantly within the actual range of content. In the case of double-measurements of m samples it holds that m 0 00 2 (yj yj ) j=1 s = (4.14) y 2 m

0 00 where yj and yj are connected values of double-measurements. The number of degrees of freedom is df = m. Relative standard deviation syrel , srel,orrsd(y )

sy s = (4.15) yrel y

as a measure that is independent of the magnitude of the observation (the term \coefˇcient of variation" instead of relative standard deviation is not recommended by IUPAC, see IUPAC Orange Book [1997, 2000]) Conˇdence intervals { measured values, following a Gaussian distribution, see Eq. (4.6a), may occur principally in the whole range of deˇnition, 1

Fig. 4.5. Integration intervals of the Gaussian distribution

Table 4.1. Corresponding integration intervals and levels of signiˇcance

˙ : P =0:683 ˙ 1:96 : P =0:95 ˙ 2 : P =0:955 ˙ 2:58 : P =0:99 ˙ 3 : P =0:907 ˙ 3:29 : P =0:999

In the case of ˇnite sample size in analytical practice, the quantiles of Stu- dent's t-distribution are used as realistic limits. The distance between a mean y and its conˇdence limits is calculated according to

sy t1˛; y = s t = p (4.16) y 1 ˛; n with sy being the standard deviation of a measurement series of n individual values from which y has been determined; is the number of degrees of freedom. The level of signiˇcance P =1 ˛ has explicitly to be ˇxed. The total (two-sided) conˇdence interval includes the mean and is given in the form cnf(y )=y ˙ y (4.17) In general, this interval includes P 100% of all measured values, i.e., in the case of n = 20 individual measurements, one value outside the conˇdence interval corresponds to the statistical expectation for P =1 ˛ =0:95. One-sided conˇdence intervals cnf(y )=y + y and cnf(y )=y y , respectively, are of importance for the control of limiting values and for test statistics. Of interest for analytical chemistry are at least two further distributions, the logarithmic normal distribution for analytical results at the trace- and ultra-trace level, and the Poisson distribution for discrete results (e.g., counts of impulse summator in XRF). In the ˇrst case it is not the measured values themselves that are normally distributed but their logarithms. Consequently, the parameters of logarithmic-normal-distributed results are estimated as geometric mean 74 4 Statistical Evaluation of Analytical Results n n 1 y = n y = antilg lg y (4.18) geom i n i i=1 i=1 and the variance of the logarithms n 2 n n lgyi 2 2 i=1 (lg yi lg y ) (lg yi) n s2 = i=1 = i=1 (4.19) lg y n 1 n 1 Because this is an unwieldy quantity, for the characterization of the unsymmetrical scattering of the results the dispersion factor (scattering factor) is used:

slgy = antilg slgy =10 (4.20) In case of unsymmetric distributed measured values, the dispersion factor can be used to estimate a relative dispersion measure that has the character of rsd(y ):

s(lgy )rel = 1 (4.21) Sometimes there exists doubt about normal and log-normal distribution and the actual character of the distribution is unknown. Then so-called \distribution-free"(robust) parameters can be applied. The mostly used of them is the (common) median1 y(n+1)=2 if n is odd medfyig = (4.22) (yn=2 + yn=2+1)=2ifn is even For this type of estimation the results of the measurements have to be arranged in numerical order. Together with the median also robust measures of dispersion, such as median absolute deviation, madfyig = medfjyi medfyigjg, and interquantile ranges (Sachs [1992]) should be used. In case of unsymmetric distributions both geometric mean and median are smaller than the arithmetic mean. In the same way as the distribution converges towards a normal one, geometric mean and median turn into the arithmetic mean. Poisson The discrete distribution is only characterized byp one parameter, the mean Y . The standard deviationp is given by sY = Y and the relative standard deviation by sYrel =1= Y . In general, the total variation of a measured result y is composed of several variation components y1;y2;:::;ym. In the course of the analytical process, all the steps of the analytical procedure (e.g., sampling, sample preparation, separation, and measurement) and of single operations (e.g.,

1 There exist a large number of \medians" on which an overwiew can be found, e.g., in Danzer [1989]; Huber [1981]; Hampel et al. [1986]; Rousseeuw and Leroy [1987]. 4.2 Uncertainty Concept 75 difference and comparative measurements) contribute to the total error. The combination of the individual errors is determined partly by statistics and partly by functional relationships of the form y = f (y1;y2;:::;ym). In the case of independent variables y1;y2;:::;ym, the total error can be estimated according to the Gaussian law of error propagation @f 2 @f 2 @f 2 2 2 2 2 y = y1 + y2 + + ym (4.23) @y1 @y2 @ym whereby it follows for the simplest and most often used relations: y = y1 + y2 2 2 2 y = y + y (4.24a) y = y1 y2 1 2 ⎫ y = y1 y2 ⎬ 2 2 2 y y1 y2 y1 ⎭ = + (4.24b) y = y y1 y2 y2 as well as Eq. (4.11). In the case of difference measurements of values of equalp order , e.g. for blank correction, the total error is mostly given by

y = 2y1 because of y1 y2 . In case of correlated parameters, the corresponding covariances have to be considered. For example, correlated quantities occur in regression and calibration (for the difference between them see Chap. 6), where the coefˇcients of the linear model y = a + b x show a negative mutual dependence.

4.2 Uncertainty Concept Traditionally, analytical chemists and physicists have treated uncertainties of measurements in slightly different ways. Whereas chemists have ori- ented towards classical error theory and used their statistics (Kaiser [1936]; Kaiser and Specker [1956]), physicists commonly use empirical uncertainties (from knowledge and experience) which are consequently added according to the law of error propagation. Both ways are combined in the modern uncertainty concept. Uncertainty of measurement is deˇned as \Parameter, associated with the result of a measurement that characterizes the dispersion of the values that could reasonably be attributed to the measurand" (ISO 3534-1 [1993]; EURACHEM [1995]). Such a parameter may be, e.g., standard deviation, or a given multiple of it, or a one-sided conˇdence interval attributed to a ˇxed level of conˇdence. In general, uncertainty of measurement comprises many components. These uncertainty components are subdivided into (i) Such that may be evaluated from the statistical distribution of the results of series of measurements and can be characterized by experimental standard deviations. 76 4 Statistical Evaluation of Analytical Results (ii) Such that can be evaluated from assumed probability distributions based on experience or other information (these components can also be characterized by standard deviations or corresponding parameters).

Occasionally, both these uncertainty components are denoted (i) as \type A"- and (ii) as \type B" uncertainties. It is an important fact, that \it is understood that the result of the measurement is the best estimate of the value of the measurand, and that all components of uncertainty, including those arising from systematic effects, such as components associated with corrections and reference standards, contribute to the dispersion" (ISO 3534-1 [1993]). Therefore, uncertainty marks the limits within which a result is accurate, i.e. precise and true (Fleming et al. [1996]). Uncertainty is estimated in various steps which can be summarized as shown in Fig. 4.6.

Fig. 4.6. The uncertainty estimation process, according to EURACHEM [1995]; Fleming et al. [1996]

At ˇrst, a clear statement should be made of the measured value y and which relationship exists between y and the parameters p1;p2;:::;pm on which it depends. If possible, that should be done in form of a mathematical equation, y = f (p1;p2;:::;pm). From this the sources of uncertainty for each part of the process should be derived and listed. Some of the parameters on their part depend from other variables pij. Also these dependencies have to be considered in form of equations or schemes, where pictograms, spreadsheets, and cause-effect diagrams (as schematically shown in Fig. 4.7) may be applied as useful tools. The quantiˇcation should start with a rough estimation of the order of magnitude of each uncertainty contribution pi and pij. Insigniˇcant one can be neglected because the uncertainty components are added according to a squared model. The signiˇcant values should be reˇned in subsequent stages and converted to parameters u(pi) which correspond to standard deviations. In the case that the parameters pi are independent from each other, the combined uncertainty is given by ! u y (p1;p2;:::) = " 2 ! 2 ! (4.25) @y 2 @y 2 u(p1) + u(p2) + @p1 @p2 4.2 Uncertainty Concept 77

Fig. 4.7. Diagram of primary and secondary effects of parameters pi and pij on the measured value y

If parameters are not independent but correlated with each other, then the relationship is more complex because the covariance has to be considered: ! u y (p1;p2; ;pm) = 2 ! 2 ! m ! (4.26) @y 2 @y 2 @y @y u(p1) + u(p2) + s pi;pj @p1 @p2 @pi @pj i;j=1

The terms s(pi;pj ) are the covariances statistically deˇned by

s(pi;pj )=s(pi) s(pj ) r (pi;pj ) (4.27a) and in connection with the combined uncertainty by

s(pi;pj )=u(pi;pj )=u(pi) u(pj ) r (pi;pj ) (4.27b) According to Eq.(4.27a,b), the covariance is composed of the product of the standard deviations or uncertainties, respectively, of the parameters pi andpj and the correlation coefˇcient between them: n

(pik p i)(pjk p j ) " k=1 r (pi;pj )= (4.28) n n 2 2 (pik p i) (pjk p j ) k=1 k=1 which is deˇned in the range 1 r (pi;pj ) +1. If values of r (pi;pj ) approximate to +1 or 1, then a strong correlation (positive or negative, respectively) is indicated. On the other hand, an approximation to 0 indicates missing correlation. From the combined uncertainty the extended uncertainty is calculated. The extended combined uncertainty U (y ) represents an interval that con- 78 4 Statistical Evaluation of Analytical Results tains a (sufˇciently) large part of the values of a series of measurements that could reasonably be attributed to the measurand U (y )=k u(y ) (4.29) ! in which u(y )=u y (p1;p2; ;pm) as calculated according to Eqs. (4.25) or (4.26), respectively. The coverage factor usually is chosen k =2:::3. Table 4.1 shows that in case of normally distributed measured values, k =2 covers an interval in which 95.5% of the values are found (k = 3 correspondingly 99.7%). If the type of distribution is unknown, k = 2 does not guarantee a sufˇcient level of conˇdence. The uncertainty interval includes the estimated mean y and is given by unc(y )=y ˙ U (y ) (4.30) In general, the uncertainties of measured values u(y ) are converted into that of analytical values u(x) by means of the sensitivity S and its uncertainty u(S): u(y;S) u(x)= : (4.31) S Therefore, the uncertainty interval of x are given by unc(y;S) y ˙ U (y;S) unc(x)=x ˙ U (x)= = (4.32) S S The signiˇcance of the uncertainty concept in analytical chemistry has increased in the last century, notwithstanding that at ˇrst some conformity was missed. But inconsistencies have been dispelled (see Thompson [1995]; AMC [1995]) and operational approaches have been presented by Hund et al. [2001]. Numerous examples of application have been given in EURACHEM [1995]. The uncertainty concept is composed of both chemists' and physicists' approaches of handling of random deviations and substitutes so classical error theories in an advantageous way.

4.3 Statistical Tests

Statistical tests make it possible, objectively to compare and interpret experimental data. They base on a test statistic to verify a statistical hypothesis about a: (i) Parameter (ii) Distribution (iii) Goodness of ˇt Generalizations which go beyond the given data are normally not possible. Statistical tests can be carried out with measured values, y , and analytical values, x, respectively, if there exists a linear relationship between 4.3 Statistical Tests 79 them. If this not apply, it depends on the problem, which of the measures has to be tested2, y or x.

4.3.1 Null Hypotheses

Statistical tests are based on hypotheses, so-called null hypotheses H0, the statements of which are veriˇed by statistical tests. In this context, the following rules apply: (i) Null hypotheses must always be formulated in an afˇrmative way, i.e., H0 : 1 = 2 (two subsets with the means x1 and x2 belong to the same 2 2 population 1 = 2), or H0 : 1 = 2 (the variances of two subsets are identical).

(ii) Every null hypothesis H0 has an alternative hypothesis HA which is conˇrmed if the null hypothesis is rejected, i.e., if the test leads to a negative result, e.g., HA : 1 =/ 2 (the compared means x1 and x2 differ signiˇcantly, and, thus, belong to different populations). (iii) A non-rejection of a null hypothesis does not mean its acceptance. If a test does not result in a signiˇcant difference of two compared parameters, this merely means, that for reasons of the existing data, the difference are not conclusive. Evidence for correspondence is not provided. Such evidence can only be conˇrmed indirectly, see e.g., Doerffel [1990]; Doerffel et al. [1990]. (iv) Each test result is only valid for a certain (freely chosen) level of sig- niˇcance P , underlying the test procedure. Thus, a test result carries a risk of ˛ =1 P . In general, a level of signiˇcance P =0:95 is chosen (corresponding to ˛ =0:05). In cases where great signiˇcance or consequence is attached to the test result, a higher level of signiˇcance (P =0:99) has to be chosen. The following decisions have to be made:

{ H0 for P =0:95 not rejected: the difference is considered to be non conclusive.

{ H0 for P =0:95 rejected: the difference is regarded as guaranteed in the normal case.

{ H0 for P =0:99 rejected: the difference is highly signiˇcant. (v) Every statistical test may possibly result in two different kinds of error, as shown in Table 4.2, i.e., { To reject the null hypothesis erroneously although it is true (error of ˇrst kind, false-negative, risk ˛). { Not to reject the null hypothesis by erroneously though the alternative hypothesis is true (error of second kind, false-positive, risk ˇ).

2 Therefore, both quantities will be used in the following to formulate the tests 80 4 Statistical Evaluation of Analytical Results

Table 4.2. Types of errors for statistical tests of null hypotheses PP PP H0 is PP H0 is PP True False by the test PPP Error of second kind Not rejected Test result OK \consumer risk" \false alarm" Error of ˇrst kind Rejected \producer risk" Test result OK \false alarm"

Means and standard deviations calculated according to Eqs. (4.9) and (4.12) or the following, respectively, are only characteristic for sample subsets if certain pre-requisites are satisˇed, the fulˇlment of these can be veriˇed by tests. The observed values of measurement series have to (i) Be normally distributed (ii) Vary randomly and show no systematic trend (iii) Be free of outliers This, on the one hand, is in order to determine correct estimates of the mean and the variation of the measurement series, and, on the other hand, to allow the statistical comparison of these parameters to those of other measurement series.

4.3.2 Test for Measurement Series (1) Rapid test for normal distribution: Range test of David et al. [1954] R q^ = (4.33) R s with R = ymax ymin being the range and s being the standard deviation. If the calculated test value q^R is not within the limits of the tabulated values (which can be found, e.g., by Sachs [1992]; Graf et al. [1987]), a normal distribution of the measured values may not be assumed. Then, it can be tested if a normal distribution is obtained after transformation of the measured values, e.g., a logarithmic one. If there is still no success, the measurement series should be evaluated by methods of robust statistics (see e.g. Danzer [1989]). The existence of a normal distribution can only be conˇrmed by a goodness-of-ˇt test (e.g., 2, according to Kolmogoroff [1941] and Smirnoff [1948]).

(2) Test for a trend

For a measurement series y1, y2,...,yn (in the sequence of the measurement, not classiˇed) the test parameter 4.3 Statistical Tests 81

n 2 (y y ) 2 = i 1 i (4.34) n 1 i=2 has to be formed (von Neumann et al. [1941]). The measured values can be regarded as independent and, therefore, as varying at random for 2 2s2. A trend has to be considered if 2 < 2s2. For an exact test 2=s2 has to be calculated and compared to the limits given e.g. in Sachs [1992].

(3) Test for outliers Among numerous tests for outliers presented in the literature, in analytical practice the following have turned out to be especially useful: a) For measurement series of limited size (n 25): Test for outliers according to Dean and Dixon [1951]; Dixon [1953]: The measured values are sorted in ascending or descending order, depending on whether the suspected outlier value y1 deviates to higher or lower values. The test statistic is formed depending on the data set # # #y y # ^ # 1 b # Q = # # (4.35) y1 yk where indices b and k may have the following values depending on the measurement series (n is the number of measured values): b = 2 for 3 n 10;b= 3 for 11 n 25 k = n for 3 n 7;k= n 1 for 8 n 13; k = n 2 for 14 n 25 The comparison is done with the signiˇcance limits Q(n;˛), given, e.g. in Sachs [1992]; Graf et al. [1987]. b) For nearly any measurement series (3 n 150): Test for outliers according to Grubbs [1969]: # # #y y # G^ = (4.36) s The test statistic is compared with the signiˇcance limits G(n;˛), given e.g. in Graf et al. [1987].

4.3.3 Comparison of Standard Deviations

Two standard deviations, s1 and s2, with corresponding degrees of freedom, 1 and 2, are compared by means of the F -test. The test statistic s2 ^ 1 F = 2 (4.37) s2 82 4 Statistical Evaluation of Analytical Results normally with s1 >s2, is compared to the corresponding quantile of the F -distribution; see Doerffel [1990] Sachs [1992]; Graf et al. [1987]. For ^ F>F1˛;1;2 it is proven that s1 is signiˇcantly larger than s2. Comparison of Several Standard Deviations:

(1) For measurement series of equal size (n1 = n2 = n3=...): Hartley test (Fmax-test, Hartley [1950]): s2 ^ max 0 Fmax = 2 (4.37 ) smin Tables with the signiˇcance limits are given in Sachs [1992]; Graf et al. [1987]. (2) Cochran test (Cochran [1941]):

2 ^ smax Gmax = 2 2 2 (4.38) s1 + s2 + + sk preferentially applied to compare k samples if one of the variances is essentially larger than all others. Tables are given in Sachs [1992]; Graf et al. [1987]. (3) Bartlett test (Bartlett [1937a, b]): k 2:3026 ^2 = lg s2 lg s2 (4.39) c i i i=1 with = n k = i being the total number of degrees of freedom 2 2 2 (k number of groups), s = isi = being the weighted variance, si being the variances of the i-th group with degrees of freedom i and

k 1 1 i c =1+i=1 (4.40) 3(k 1) which has the character of a correction factor; if the number of degrees 2 of freedom i is not too small, c approximates 1. The test statistic will be compared to the corresponding quantile of the 2-distribution 2 Doerffel Sachs Graf ˛; ; see [1990]; [1992]; et al. [1987].

4.3.4 Comparison of Measured Values

(1) Comparison of two means y 1 and y 2 of different measurement series with n1 resp. n2 measurements having the standard deviations s1 resp. s2 is done by the t-test: 4.3 Statistical Tests 83 # # # # $ y y n n t^ = 1 2 1 2 (4.41) sav n1 + n2

with the (weighted) average standard deviation " 2 2 (n1 1)s1 +(n2 1)s2 sav = (4.42) n1 + n2 2

The t-test in this form can only be applied under the condition that 2 2 the variances of the two sample subsets, s1 and s2 , do not differ signiˇ- cantly. This has to be checked by the F -test beforehand. The test statistic t^ has to be compared to the related quantile of the t-distribution t1˛; where = n1 + n2 2. 2 2 If the variances s1 and s2 , differ signiˇcantly, a generalized t-test (tW test) according to Welch [1937] can be applied: # # #y y # %1 2 tW = (4.43) s2 s2 1 + 2 n1 n2

The comparison is done again with t1˛; , but in this case it is 2 s2 s2 1 + 2 n1 n2 = 2 2 (4.44) s2 s2 1 2 n n 1 + 2 n11 n21

If more than two means have to be compared, the t-test cannot be applied in a multiple way. Instead of this, an indirect comparison by analysis of variance (ANOVA) has to be used, see (3) below. (2) Comparison of an experimental mean with a prescribed resp. true value: simpliˇed t-test: The decision, if a mean of a sample, y , differs randomly or signiˇcantly from a prescribed mean, , leads to the question, if the conˇdence interval of the experimental mean,# cnf(y#), includes the mean or # # not, i.e., is the absolutep difference y smaller or larger than the interval: y = s t1˛; = n: # # #y #p t^ = n (4.45) s 84 4 Statistical Evaluation of Analytical Results (3) Comparison of several means: analysis of variance (ANOVA): Because no direct way exists to test simultaneously more than two means for signiˇcant differences, the comparison of several means, y 1; y 2;:::;y k, is traced back to the comparison of variances. One-way analysis of variance is based on a linear model like the following:

yi = y + ai + ei (4.46)

where three different models have to be distinguished (Eisenhart [1947]): (i) Model I with ˇxed effects (model \ˇxed") which is suitable for comparisons of means on the basis of the null hypothesis H0: 1 = 2 = :::= k (ii) Model II with random effects (model \random") that bases on the 2 null hypothesis H0: a =0(\proper" analysis of variance) (iii) Model \mixed"

Regarding the comparison of means, ai and ei can be interpreted as to correspond to the deviations between the measurement series and within 2 2 them, between and within, respectively. Therefore, the comparison is carried out according to

k ! 2 1 n y y 2 k1 i i F^ = between = i=1 (4.47) 2 k within 1 2 nk si (n 1) i=1 with k number of means to be compared, y i mean of the i-th measurement series with ni individual values, si standard deviation of the i-th measurement series, n = ni total number of all individual measurements, 1 y = n niy i weighted total mean. The degrees of freedom are 1 and 2. ^ F exceeds the corresponding quantile of the F -distribution F1˛;1;2 if at least one of the means differs signiˇcantly from the others. This global statement of variance analysis may be speciˇed in the way to detect which of the mean(s) differ(s) from the others. This can be done by pairwise multiple comparisons (Tukey [1949]: Games and Howell [1976]: see Sachs [1992]). In the simplest case, if all sample sub groups have the same size, n1 = n2 = :::= nk, Dixon's test for outliers can be used (see Sect. 4.3.2). Then, in Eq. (4.35), instead of the individual values, the means are entered. 4.4 Reliability of Qualitative Analytical Tests 85 4.4 Reliability of Qualitative Analytical Tests

In principle, there is no fundamental contrast between qualitative, (semi- quantitative), and quantitative analyses. The analytical signal is generated in the same way, only the detection and evaluation is done on the basis of a more rough scale, in qualitative analysis only in form of a yes/no decision. Today an increasing importance of qualitative analysis can be stated in certain ˇelds. This is due to an increasing number of materials under study, especially active agents of interest on the one hand, and the many and diverse ways of synthesis (e.g., by combinatorial chemistry) on the other hand as well as the increasing demands on quality. Because analytical laboratories in research and routine control would be overtaxed in their capacity if full quantitative analyses were done generally, screening methods become more and more signiˇcant. Screening techniques can be understood as to be ˇltering procedures of samples. The principle of screening consists in giving an overview on constituents in certain samples, namely: (i) Whether a threshold value (speciˇcation limit) is exceeded or not (ii) Whether constituents are present in addition to such that are well- known, in general, to specify the type of the sample (e.g., identiˇcation of the type of steel by recognizing typical alloying constituents additionally to that always present like C, S, P, Si, and Mn) Although the term \screening methods" is often used as a synonym for qualitative analysis, it is mostly related to (i) the scheme of which is given in Fig. 4.8; see also Trullols et al. [2004].

Samples Screening + +++ Further procedures, e.g. technique: + + + quantitative analysis + ––+ – + – + – – + – + – + – Discrimination yi ≤ ySL – – – – – – – – – yi > ySL No further procedures

Fig. 4.8. Principle of screening techniques. + samples for which yi >ySL, samples for which yi ySL; ySL is the speciˇcation limit

Screening methods may be classiˇed according to the detection of signals into: (a) Sensorial detection (mostly visibility) traced back to changes in absorption or transmission (e.g., color of solution or test strips, turbidity). (b) Instrumental detection, where in principle any instrumental technique may be used. Screening by instrumental analysis can be done by means 86 4 Statistical Evaluation of Analytical Results of reduced calibration expense where only a comparison with a reference sample containing the analyte at the speciˇcation level, xSL, giving so a response ySL, is carried out. According to the practical equipment there are useful tools, so-called test kits, which are units that contain all the reagents and a simple instrumentation in form of plates, tubes and wells. The test kits work rapidly, are easily to handle and ˇeld-portable. Frequently, biochemical principles are applied, especially immunoassay techniques which use body-antibody reactions. The discrimination limit in screening may have different character. It may be a detection limit or a pre-set value set by a responsible authority, e.g., an ofˇcial agency, medical institutions, or clients. Although screening tests are evaluated qualitatively, as a rule, quantitative aspects of test statistics and probability theory have to be considered. In this respect, validation of qualitative analytical procedures has been included in international programs and concepts, see Trullols et al. [2004]. The application of screening methods requires that proper reference samples are available. These must contain the analyte exactly at or close to the speciˇcation level. Such reference samples have to be veriˇed by quantitative analysis using reference methods. The reliability of screening methods is usually expressed in terms of probability theory. In this regard, the conditional probability, P (BjA), characterizing the probability of an event B given that another event A occurs, plays an important role. (1) True positive rate3: probability that the test result is positive in case that the analyte A is present (above the speciˇcation limit) in the test sample

tp TPR = P (T + jA)= (4.48) tp + fn

where tp are the true positives and fn the false negatives. (2) True negative rate: probability that the test result is negative when the analyte is not present, A (i.e. equal or below the speciˇcation limit, respectively)

# tn TNR = P (T #A)= (4.49) tn + fp

where tn are the true negatives and fp the false positives.

3 In clinical chemistry and medical diagnostics the true positive rate is called \sensitivity rate" and the true negative rate \speciˇcity rate"(O'Rangers and Condon [2000]) without any relation to the general deˇnition of the terms sensitivity and speciˇcity and their use in analytical chemistry (see Sects. 7.2 and 7.3). 4.4 Reliability of Qualitative Analytical Tests 87 (3) False positive rate: probability that the test result is positive when the analyte is not present, A

# fp FPR = P (T + #A)= (4.50) tn + fp

(4) False negative rate: probability that the test result is negative when the analyte is present

fn FNR = P (T jA)= (4.51) tp + fn

Some other important quantities in test theory are: (5) Prevalency: probability that A is present in n samples:

tp + fn PV = P (A)= (4.52) n where n = tp + tn + fp + fn is the total number of samples (tests). (6) Concordancy rate: proportion of correct test results:

tp + tn CR = (4.53a) n where CR 1. The correctness of test is also expressed by (7) Test efˇciency: # TE = P (A #T + )+P (A jT ) 2 (4.53b)

The problem of test efˇciency and concordance may be illustrated by a scheme (Table 4.3) which corresponds to that given in Table 4.2.

Table 4.3. Scheme of test results for screening procedures; tp true positive, fp false positive, tn true negative, fn false negative, n total number of tests XX XX Test result XX Positive (T +) Negative (T ) Sum of situations Situation XXX A is present (A) tp fn tp + fn Error of second kind

A is not present (A) fp tn tn + fp Error of ˇrst kind Sum of test results tp + fp tn + fn n = tp + tn + fp + fn 88 4 Statistical Evaluation of Analytical Results From the decision variants and probabilities given above, the prediction values as applied in Eq. (4.53b) play an important role in decision theory, namely: { Positive prediction rate: proportion of samples which are positively tested when the analyte A is present in the sample # P (A) P (T + jA) PPR= P (A #T + )= # P (A) P (T + jA)+[1 P (A)] [1 P (T #A)] (4.54) P (A) P (T + jA) = # P (A) P (T + jA)+P (A) P (T + #A) and { Negative prediction rate: proportion of samples which are negatively testes if the analyte is not present (A) in the sample # P (A) P (T #A) NPR = P (A jT )= # P (A) P (T #A)+P (A) [1 P (T + jA)] # (4.55) P (A) P (T #A) = # P (A) P (T #A)+P (A) P (T jA)

False positive and false negative decisions result from unreliabilities that can be attributed to uncertainties of quantitative tests. According to Fig. 4.9 the belonging of test values to the distributions p(yLSP )orp(ySCR), respectively, may be affected by the risks of error ˛ and ˇ (see Sect. 4.3.1) which corresponds to false positive (˛) and false negative (ˇ) test results.

Fig. 4.9. Distributions of measured values which belong to the content at the limit of speciˇcation xLSP and at the screening limit xSCR;yDIS is the limit of discrimination 4.4 Reliability of Qualitative Analytical Tests 89

Fig. 4.10. Performance characteristic curves of screening tests of binary classiˇcations: a ideal curve shape, b experimentally obtained curve following Simonet et al. [2004]; PPR is the probability of obtaining positive responses, TPR true positive rate (Eq. 4.48), TNR true negative rate (Eq. 4.49), FPR false positive rate (Eq. 4.50), FNR false negative rate (Eq. 4.51), xl and xu are the lower and upper limits of the unreliability region

Another way to characterize the performance of screening methods for binary classiˇcation is the construction of performance curves. A performance characteristic curve (PCC) represents the probability of positive test results vs the concentration of the analyte as shown in Fig. 4.10. The performance curve presents graphically the relationship between the probability of obtaining positive results PPR, i.e. xi xLSP on the one hand and the content x within a region around the limit of discrimination xDIS on the other. For its construction there must be carried out a larger number of tests (n 30) with samples of well-known content (as a rule realized by doped blank samples). As a result, curves such as shown in Fig. 4.10 will be obtained, where Fig. 4.10a shows the ideal shape that can only be imagined theoretically if \inˇnitely exact" decisions, corresponding to measured values characterized by an \inˇnitely small" conˇdence interval, exist. The curve illustrates the sharpness of tests depending on the discrimination limit. In this way, TPR and TNR may be recognized and the unreliability region around the limit of speciˇcation can be estimated. Beyond the limits of the unreliability interval, it is possible to classify samples correctly apart 90 4 Statistical Evaluation of Analytical Results from a deˇned risk of error (Pulido et al. [2002]; Simonet et al. [2004]; Trullols et al. [2004]).

4.5 Statistical Quality Control

The principles of quality assurance are commonly related to product and process control in manufacturing. Today the ˇeld of application greatly ex- panded to include environmental protection and quality control within analytical chemistry itself, i.e., the quality assurance of analytical measurements. In any ˇeld, features of quality cannot be reproduced with any absolute degree of precision but only within certain limits of tolerance. These depend on the uncertainties of both the process under control and the test procedure and additionally from the expense of testing and controlling that may be economically justiˇable.

4.5.1 Quality Criteria for Analytical Results

Analytical methods, particularly those used by accredited laboratories, have to be validated according to ofˇcial rules and regulations to characterize objectively their reliability in any special ˇeld of application (Wegscheider [1996]; EURACHEM/WELAC [1993]). Validation has to control the performance characteristics of analytical procedures (see Chap. 7) such as accuracy, precision, sensitivity, selectivity, speciˇcity, robustness, ruggedness, and limit values (e.g., limit of detection, limit of quantitation). Within the scope of quality agreements in production, environment, or laboratory, quality is often stipulated to a standard value x0 (target value). For demands regarding quality, this standard value may be an upper limit (e.g. in case of pollution and contamination) or a lower limit (e.g. for active reagents). The statistical situation is the same when a quality criterion has to exceed or fall below a standard value. The problem is illustrated here by the practical situation of manufacturer and customer as shown in Fig. 4.11. If an analytical test results in a lower value xi x0, then the customer may reject the product as to be defective. Due to the variation in the results of analyses and their evaluation by means of statistical tests, however, a product of good quality may be rejected or a defective product may be approved according to the facts shown in Table 4.2 (see Sect. 4.3.1). Therefore, manufacturer and customer have to agree upon statistical limits (critical values) which minimize false-negative decisions (errors of the ˇrst kind which characterize the manufacturer risk) and false-positive decisions (errors of the second kind which represent the customer risk) as well as test expenditure. In principle, analytical precision and statistical security can be increased almost to an unlimited extent but this would be reected by high costs for both manufacturers and customers. 4.5 Statistical Quality Control 91

Fig. 4.11. Distributions of analytical results for quality assurance; C is the distribution of analytical results at the critical customer limit, M at the critical manufacturer limit, SV is the distribution of the standard value

With the aid of consequences resulting from wrong decisions, risks R can be given for the manufacturer M as well as for the customer C:

RM = w ˛ FM (4.56a)

RC =(1 w ) ˇ FC (4.56b) with w = P (x = xcrit(M)) representing the a priori probability that a determined quality value x is equal to the critical manufacturer limit and (1 w )=P (x = xcrit(C)) that is equal to the critical customer limit. The parameters FM and FC characterize consequences (e.g. costs) resulting from wrong decision for the manufacturer and customer, respectively; ˛ is the (one-sided) manufacturer risk (error of the ˇrst kind) and ˇ the (one-sided) customer risk (error of the second kind). The relationship between the target value x0 and the distribution of the critical values are illustrated in Fig. 4.11 for manufacturer (M) and customer (C). For x>x0 the quality is better than agreed while x

The critical values xcrit(C) and xcrit(M) being tolerance limits are derived as conˇdence limits of x0. They are given by p xcrit(M) = x0 + st1˛; = n (4.57a) for the upper tolerance limit with the standard deviation s of the analytical procedure and n parallel determinations, as well as p xcrit(C) = x0 st1ˇ; = n (4.57b) for the lower tolerance limit. If manufacturer and customer have agreed upon x0 not to be a target value but a guarantee value, x0;guar, with ˇxed signiˇcance level P =1 ˛, then that has to supply at least the quality xcrit(M). The statement whether an analytical value x = x0;guar is still accepted (x x0;guar) or not (x>x0;guar) is then also part of the agreement. In these cases, the customer's risk is that in ˛ 100% of all cases the quality can be worse than x0. Quality limits are often stipulated by laws, regulations, or standards; however, they are frequently determined by mutual agreements, too. Con- sidering the particular risks RM resp. RC, especially the values ˛ and ˇ have to be reconciled, whereby, in practice, ˛ = ˇ is often taken as the basis.

4.5.2 Attribute Testing

In contrast to variable testing (comparison of measured values or analytical values), attribute testing means testing of product or process quality (non- conformity test, good-bad test) by samples. Important parameters are the sample size n (the number of units within the random sample) as well as the acceptance criterion naccept, both of which are determined according to the lot size, N , and the proportion of defective items, p, within the lot, namely by the related distribution function or by operational characteristics. According to the number of defective items ndefect, which have been determined in the sample results are required for,:

ndefect naccept acceptance of the test items

ndefect >naccept rejection : Attribute testing can also be effected on the basis of two or more (m) samples. The disadvantage for the higher expenditure to determine the sample sizes n1;n2;:::;nm, the acceptance and rejection rates naccept;1;naccept;2;:::;naccept;m resp. nrejet;1;nreject;2;:::;nreject;m must be set against the advantage of the possibility of coming to unambiguous decisions in clearly deˇned situations with only a ˇrst small random sample i =1. In contrast to classical statistical tests, three test outcomes exist:

(i) ndefect;i naccept;i acceptance

(ii) ndefect;i >naccept;i rejection 4.5 Statistical Quality Control 93

(iii) naccept;i

4.5.3 Sequential Analysis

The principle of sequential analysis consists of the fact that, when comparing two different populations A and B with pre-set probabilities of risks of error, ˛ and ˇ, just as many items (individual samples) are examined as necessary for decision making. Thus the sample size n itself becomes a random variable. Sequential investigations may be used for both attribute testing and quantitative measurements (variable testing). The fact that it is enough to perform only as many tests or measurements as are absolutely necessary, is of great advantage in such cases if individual samples are either difˇcult to obtain or expensive, or if the same is true for the measurement. Based on the actual result of every individual measurement, it has to be checked whether a decision may now be taken or whether studies have to be continued. According to the ˇnal aim of sequential analysis, one can differentiate between closed sequential test plans that always come to a decision A > BorA< B (possibly with a large test expenditure) and open sequential test plans, which also allow { after a certain test expenditure { the statement A = B. Data evaluation can be done by calculation or graphically. An example of graphical sequential analysis is given in Fig. 4.12 for attribute testing. In the case of attribute testing, limit curves for acceptance and rejection are normally straight lines. In case of variable testing, they are mostly nonlinear functions (Graf et al. [1987]). For variable testing, the analytical values x are represented on the ordinate axis, and a decision is taken when the acceptance or rejection curve is exceeded. Decisions to be taken after every individual sample are as follows: (a) For attribute testing:

ndefect;n naccept;n = gaccept(n) acceptance

ndefect;n nreject;n = greject(n) rejection

gaccept(n)

with ndefect;n being the number of defective items within n tested, naccept;n resp. nreject;n being the actual acceptance resp. rejection criterion for the current sample size n; gaccept(n) and greject(n) being the acceptance resp. rejection functions. 94 4 Statistical Evaluation of Analytical Results

Fig. 4.12. Graphical evaluation of sequential analysis; gaccept(n) acceptance function, greject(n) rejection function

(b) For variable testing (quantitative measurements):

xn gaccept(n;s;˛) acceptance

xn greject(n;s;ˇ) rejection

gaccept(n;s;˛)

with xn being the actual analytical value after n measurements, gaccept(n;s;˛) and greject(n;s;ˇ) being the acceptance and rejection functions (s standard deviation of the analytical procedure, and ˛ and ˇ manufacturer and customer risk, respectively).

Sequential test plans are also suitable for the direct comparison of two products or procedures based on subjective parameters as well as on measured results. This can be done without any computation, but only according to the following three criteria:

{ A is better than B { B is better than A { There is no difference between A and B

A graphical evaluation scheme of such comparisons is given in Fig. 4.13. 4.5 Statistical Quality Control 95

Fig. 4.13. Sequential test plan for direct characteristic comparison

4.5.4 Statistical Quality Control

Continuous quality control is based on principles that ˇrstly were used in the system of \quality control charts"(QCC, Shewhart [1931]). Today, admittedly the monitoring of the characteristics of a process or product in order to detect deviations from the target value is not tied to charts but is mostly done by computer, although it is frequently still called a \control chart" system. On the one hand, statistical quality control is an important tool for quality assurance within analytical chemistry itself (monitoring of test methods), and on the other for quality control of processes and products by means of analytical methods. In each case, the aim of quality control charts4 (QCC) is the representation of quality target values x0 (standard values) and their statistical limits. A control chart usually consists of ˇve lines as shown in Fig. 4.14. QCC contain { as quality target values Q { standard or reference values, x0, resp. optimum values as well as their limits. The inner pair of limits are called warning limits and the outer pair control limits (action limits). When

4 may it now represented by a paper card or a computer 96 4 Statistical Evaluation of Analytical Results

Fig. 4.14. General scheme of a quality control chart; LoS: level of signiˇcance the warning limits are exceeded, attention must be increased to recognize a possible tangency or exceeding of the control limit. The latter requires corrective action. The results of sample tests are marked in the charts as a series of points. Their sequence allows discerning quickly typical situations as some are shown in Fig. 4.15. Case A represents a normal inconspicuous process, while the other cases show out-of-control situations (OCS), namely for various reasons. Case B show the typical OCS: a value is situated outside the upper control limit and demands instantaneous action. An OCS is also indicated if a given number of successive points are above resp. below the target value (namely 7 out of 7, see case C, 10 of 11, 12 of 14, 16 out of 20). The same holds true for steadily increasing or decreasing tendencies (case D). Whereas a single crossing of the warning limit only demands closer attention, 2 out of 3 successive values exceeding the warning limit, as shown by example E, indicate OCS. The cases F, G and H illustrate exceptional situations, demanding instant attention as the process is threatening to get out of control. F shows periodic alterations, G a long-term trend, while in case H a particular great number of values are close to the control limit. According to the character of the target value Q several types of control charts can be distinguished:

{ Single value charts { Mean value charts (charts on x, median charts, blank charts) { Variation charts (standard deviation charts, s resp. srel, range charts, R) { Recovery rate charts { Cusum charts { Combination charts (e.g., charts on x s and on x R) { Correlation charts (e.g., charts on xA and xB). 4.5 Statistical Quality Control 97

Fig. 4.15. Conspicuous plots of control charts

Single value charts are only used for special purposes, e.g. as original value chard for the determination of warning and control limits or, for data analysis of time series (Shumway [1988]; Montgomery et al. [1990]). All the other types of charts are used relatively often and have their special advantages (Besterfield [1979]; Montgomery [1985]; Wheeler and Cham- bers [1990]). For the limits of means (xp= x0 = Q), the followingp bounds are chosen: UCL p= x + tP =0:99; s= n, UWL p= x + tP =0:95; s= n, LWLp= xtP =0:95; s= n,p and LCL = xtP =0:99; s= n. The limits WL = x˙2s= n and CL = x ˙ 3s= n are also often used. Parameters for the most important of the above mentioned quality control charts are given in Table 4.4. Appropriately designed, Cusum control charts give sensitive and instruc- n tive impressions on process changes. Cumulative sums Sn = i=1 (xi x0) also contain information on actual as well as on previously obtained values. Therefore their display enables one to perceive earlier changes leading to OCS than by means of the chart of original values (see Woodward and Goldsmith [1964]; Marshall [1977]; Doerffel [1990]). 98 References

Table 4.4. Control parameters for QCCs

Target value Upper limits Lower limits p p Mean value xLU = x + t˛; s= nLL = x t˛; s= n 2 2 Standard deviation sL= s n1;1˛=2 L = s n1;˛=2 U n 1 L n 1

Range RLU = DU RLL = DL R p p Recovery rate RR = ˇLU = RR + t˛; sRR= nLL = RR t˛; sRR= n (slope of the recovery function, see Eq. (4.3))

The corresponding factors of the t- and 2-distribution can be found in all textbooks on statistics, e.g., Dixon and Massey [1969]; Davies and Goldsmith [1984]; Graf et al. [1987]; Arnold [1990]; Sachs [1992]. For DU and DL see e.g. GRAF et al. [1987].

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5 This is a pseudonym of WS Gosset. 5 Studying Inuences and Optimizing Analytical Procedures

According to Sect. 3.5 the inuencing of an analytical signal by the environment of the experiment (inuence factors), i.e. interferences and other stimuli can be estimated either in a way basing on chemical facts (Eq. 3.16a) or in a statistical way (Eq. 3.16b,c). Therefore, two ways are feasible to study the signiˇcance of inuences as a requirement of a subsequent optimization.

5.1 Testing the Signiˇcance of Inuencing Factors

In analytical chemistry, the optimality criterion is frequently the relative increase of that share of the analytical gross signal that is caused by the analyte itself, namely SAAxA=yA. According to Eq. (3.16a): N m yA = yA0 + SAAxA + SAixi + IAj xj + eA0 (5.1) i=1 j=1 which is tantamount to the minimization of the interferences, SAixi, and inuencing factors, IAj xj . The quantities SAi are the cross sensitivities of the interferents i and IAj , the speciˇc strength of the inuence factors j.On the basis of this relationship the signiˇcance of interferents and factors can be studied using so-called multifactorial designs. On the other hand, Eqs. (3.16a) and (5.1), respectively, can also be interpreted as being composed of the analyte signal and diverse (more or less) anonymous deviations ei,ej or eAij , respectively, see Eq. (3.16b,c):

yA = yA0 + SAAxA + ei + ej + eA0 (5.2) On this basis, an analysis of variance (ANOVA) can be carried out to test the signiˇcance of the variations ei = SAixi, ej = IAj xj , or more in detail, eB = SABxB, eC = SAC xC etc.

5.1.1 Analysis of Variance (ANOVA) ANOVA was developed by Fisher [1925, 1935] as a statistical procedure that investigates inuences (effects) of factors on a target quantity y according to a linear model which holds in the simplest case 102 5 Studying Inuences and Optimizing Analytical Procedures

Table 5.1. Measurement and evaluation scheme of one-way ANOVA

Levels i of the factor a 1 2 ... m

Number j 1 y11 y21 ... ym1 of single 2 y12 y22 ... ym2 measurement ...... n y y ... y 1n 2n mn Sum S1 = y1j S2 = y2j ... Sm = ymj

Mean y 1 y 2 ... y m Overall mean y

Table 5.2. Variance components in one-way ANOVA

Source of Sum of squares Degrees of freedom Variance F -test variation m SS s2 Between SS = n (y y )2 = m 1 s2 = a F^ = a a i a a m 1 s2 the factor i=1 res levels m n SS Residual SS = (y y )2 = m(n 1) s2 = res res ij i res res m(n 1) (analytical i=1 j=1 random error)

Total SStotal = SSa + SSres total = m n 1

yij = y + ˛i + eij (5.3) with yij being the actual value, y the overall mean, ˛i an additive inuence of the factor a at level i, and eij the residual deviation (one-way analysis of variance, see Sect. 4.3.4). By means of ANOVA it is possible to compare both variances and means where two different models exist:

Model I (model \ˇxed") that compares means on the basis of the null hypothesis H0: 1 = 2 = :::= m and Model II (model \random") by which variances are compared on the 2 2 2 basis of the null hypothesis H0: 1 = 2 = ::: = m (corresponding to 2 H0: a =0) and additionally a mixed one (see Sect. 4.3.4 and Eisenhart [1947]). Variance analysis should advantageously be carried out on the basis of balanced experiments where the number of observations per factor level is equal (n1 = n2 = :::= nm = n). 5.1 Testing the Signiˇcance of Inuencing Factors 103 The measurement scheme of One-way analysis of variance is given in Ta- ble 5.1 for i =1:::m levels of the factor a (in analytical practice frequently a factor is studied only on two levels to compare, e.g., two laboratories, two operators, two different techniques, etc). The variance components are calculated according to Table 5.2. The estimated F^value has to be compared with the quantile of the F - distribution, F1˛; , the tables of which can be found in textbooks of statistics (e.g., Hald [1960]; Neave [1981]; Dixon and Massey [1983]; Graf et al. [1987]; Sachs [1992]). The inuence of the factor a is signiˇcant when F^ exceeds F1˛; . In case of unbalanced experiments the different size of measurement series and, therefore, degrees of freedom have to be considered as a result of which both the evaluation scheme and the variance decomposition become more complicated (see Dixon and Massey [1983]; Graf et al. [1987]). By means of Two-way ANOVA two factors can be studied simultaneously. The model

yij = y + ˛i + ˇj +(˛ˇ)ij + eij (5.4) considers the inuences ˛i of the factor a at levels i, ˇj of the factor b at levels j, and additionally the interactions (correlations), (˛ˇ)ij, of both the factors (yij actual value, y overall mean, eij residual deviation, i.e., experimental error). The scheme of measurement and evaluation of two-way ANOVA is given in Table 5.3 and the corresponding variance decomposition in Table 5.4. On the basis of two-way ANOVA two null hypotheses can be tested, namely

Ha: ˛1 = ˛2 = :::= ˛m = 0 by means of F^a compared with

F1˛;1=m1;2=(m1)(n1),

Hb : ˇ1 = ˇ2 = :::= ˇm = 0 by means of F^b compared with

F1˛;1=n1;2=(m1)(n1).

Table 5.3. Evaluation scheme of two-way ANOVA (ab model: single measurements in each point; the index point marks that levels over which is actually added up or averaged, respectively)

Levels i of the factor a Sum Mean 1 2 ... m Levels j 1 y11 y21 ... ym1 S1 = yi1 y1 of the 2 y12 y22 ... ym2 S2 = yi2 y2 factor b ...... . n y1n y2n ... ymn Sm = yim yn Sum S1 = y1j S2 = y2j ... Sm = ymj S {

Mean y1 y2 ... ym { y 104 5 Studying Inuences and Optimizing Analytical Procedures

Table 5.4. Variance components in two-way ANOVA (ab model)

Source of Sum of squares Degrees of freedom Variance F -test variation m 2 2 2 S S SS s Between SS = j = m 1 s2 = a F^ = a a n m n a a m 1 a s2 the levels i=1 res of factor a n S2 S2 SS s2 Between SS = i = n 1 s2 = b F^ = b b m m n b b n 1 b s2 the levels i=1 res of factor b m n 2 2 S 2 SStotal Total SStotal = yij total = m n1 stotal = i=1 j=1 m n m n 1 Residual SS (analytical SS = SS SS SS =(m 1)(n 1) s2 = res res total a b res res (m 1)(n 1) random error)

As a rule, interactions (˛ˇ)ij cannot be estimated if only single observations at each point of the measurement matrix are carried out. In case of single measurements, the residual error contains both experimental error and interactions, the separation of which is possible only in special cases, e.g., testing of homogeneity of solids when certain assumptions can be made. Danzer and Marx [1979] have investigated the homogeneity of steel samples by means of a destructive OES procedure. Consequently, no repeated measurements could be carried out and the residual error must be corrected by interaction terms estimated according to Mandel [1961]. If possible, two-way ANOVA should be applied doing repetitions at each level. In case of double measurements the 2ab model represented in Ta- bles 5.5 and 5.6 is taken as the basis of evaluation and variance decomposition. On the basis of this 2ab ANOVA it is possible to test three null hypotheses, namely

Ha: ˛1 = ˛2 = :::= ˛m = 0 by means of F^a compared with

F1˛;1=m1;2=(m1)(n1),

Hb : ˇ1 = ˇ2 = :::= ˇm = 0 by means of F^b compared with

F1˛;1=n1;2=(m1)(n1), and

Hab:(˛ˇ)11 =(˛ˇ)12 = :::=(˛ˇ)mm = 0 by means of F^ab compared

with F1˛;1=n1;2=mn). In the case that interactions prove to be insigniˇcant, it should be gone over to the ab model the estimations of which for the various variance components is more reliable than that of the 2ab model. A similar scheme can be used for three-way ANOVA when the factor c is varied at two levels. In the general, three-way analysis bases on block-designed experiments as shown in Fig. 5.1. 5.1 Testing the Signiˇcance of Inuencing Factors 105

Table 5.5. Evaluation scheme of two-way ANOVA (2ab model: double measurements in each point)

Levels i of the factor a Sum Mean 1 2 ... m

Levels j 1 y111, y112 y211, y212 ... ym11, ym12 S1 y1 of the 2 y121, y122 y221, y222 ... ym21, ym22 S2 y2 factor b ...... n y1n1, y1n2 y2n1, y2n2 ... ymn1, ymn2 Sm yn

Sum S1 S2 ... Sm S {

Mean y1 y2 ... ym { y

Table 5.6. Variance components in two-way ANOVA (2ab model)

Source of Sum of squares Degrees of freedom Variance F -test variation m 2 2 2 S S SS s Between SS = j = m 1 s2 = a F^ = a a 2n 2mn a a m 1 a s2 the levels i=1 res of factor a n S2 S2 SS s2 Between SS = i = n 1 s2 = b F^ = b b 2m 2mn b b n 1 b s2 the levels i=1 res of factor b SS s2 Interaction SS = SS SS SS SS =(m 1)(n 1) s2 = ab F^ = ab ab total a b res ab ab ab 2 between (m 1)(n 1) sres a and b m n 2 m n SS Residual SS = y 2 S2 m ns2 = res res ijk ij res mn (analytical i=1 j=1 k=1 i=1 j=1 random error) m n 2 2 S 2 SStotal Total SStotal = yij total =2m n 1 stotal = i=1 j=1 2mn 2mn 1

Following the scheme given in Fig. 5.1, the inuence of three factors a, band c can be studied on the basis of the linear model

yijk = y + ˛i + ˇj + k +(˛ˇ)ij +(˛)ik +(ˇ)jk +(˛ˇ)ijk + eijk (5.5)

The estimation of all the terms of Eq. (5.5) is possible for the balanced case and q repeated measurements in each cell of the data block represented in Fig. 5.1. Schemes for this and some reduced variants of three-way ANOVA are given in Scheffe [1961]; Ahrens [1967]; Dunn and Clark [1974]; Graf et al. [1987]; Sachs [1992]. By means of the three-way variance analysis according to the model at Eq. (5.5) and Table 5.7 the inuence of the factors a, b and c can be tested as well as that of the interactions, i.e. the null hypotheses: 106 5 Studying Inuences and Optimizing Analytical Procedures 2 res 2 2 2 2 2 2 2 2 2 2 ab ac bc abc a b res c res res 2 2 2 res res s res s s s s s s s s s s s s s = = = = , etc. -test = = = c F c b a bc ac ab ^ ^ ^ ^ ^ F ^ F F abc F F F ^ F 1) 1) 1) ) 1 q 1 1 1 n p p c b a ab ac bc abc res total abc (1 levels of factor 1)( 1)( SS SS SS SS SS SS SS SS SS 1)( n p m k mnpq = = mnp = = n m m Variance = ( 2 2 ( ( b c 2 2 a abc s s = s s = = = 2 total 2 res s s 2 bc 2 2 ac and the ab s s s a 1) p 1) 1) 1) 1)( 1) 1 , etc. 1 1 1 c n p n p levels of factor q i ( 1)( 1)( 1)( 1)( n p m mnpq mnp n m m m ( ( ( Degrees of freedom =( , etc. b abc levels of factor mean over the k k 2 ) i 2 2 2 ) y jk ) ) ; y b y y 2 y y 2 , and the ) levels of factor ) + k b + + k 2 2 j 2 i ) ) ) k y ijk k model with repetitions) y y j y y y y + y y y abc ij j ijkl ijkl y y j y y k ( ( i i j y y y i levels of factor + q =1 y y q y =1 l ( l ( ( j ijk p =1 y n i p =1 p =1 =1 levels of factor m =1 k k ( j i mean over the jk k k y ij i j n y y + =1 y n p =1 =1 j ( ( ( j k j npq Sum of squares mpq n m p =1 =1 =1 mnq p =1 y m =1 i j k i and the k n =1 , , the = j = = = m n a =1 a a =1 m =1 = a i b j i c m =1 SS i res SS SS pq total nq mq q SS SS = = = = ab ac bc SS abc SS SS SS levels of factor levels of factor levels of factor i i i a b c c Variance components in three-way ANOVA ( b c c and total mean mean over the mean over the mean over the b and and and , Source of variation Between the levels of factor Between the levels of factor Between the levels of factor Interaction between a Interaction between a Interaction between b Interaction between a Residual error Total i ij ijk Table 5.7. y y y y 5.1 Testing the Signiˇcance of Inuencing Factors 107

Fig. 5.1. Principle of three-way ANOVA: data are arranged in rows, columns, and layers (some examples of data pixels are given)

1 Ha: ˛1 = ˛2 = :::= ˛m = 0 by means of F^a comp ! F1˛;1=m1;2=mnp1, ^ Hb : ˇ1 = ˇ2 = :::= ˇm = 0 by means of Fb comp F1˛;1=n1;2=mnp1, ! ^ Hc: 1 = 2 = :::= m = 0 by means of Fc comp F1˛;1=p 1;2=mnp1, !

Hab:(˛ˇ)11 =(˛ˇ)12 = :::=(˛ˇ)mn = 0 by means of F^ab comp ! F1˛;1=(m1)(n1);2=mnp1),

Hac:(˛)11 =(˛)12 = :::=(˛)mp = 0 by means of F^ac comp ! F1˛;1=(m1)(p 1);2=mnp1),

Hbc:(ˇ)11 =(ˇ)12 = :::=(ˇ)np = 0 by means of F^ab comp !

F1˛;1=(n1)(p 1);2=mnp1), and

Habc:(˛ˇ)111 =(˛ˇ)121 = :::=(˛ˇ)mnp =0byF^ab comp ! F1˛;1=(m1)(n1)(p 1);2=mnp1).

1 comp symbolizes \compared with" ! 108 5 Studying Inuences and Optimizing Analytical Procedures In some cases interactions are improbable and information on them is not needed. Then reduced variants of three-way ANOVA can be applied by which the effects of the main factors can be estimated more reliable (see Dunn and Clark [1974]; Graf et al. [1987]; Sachs [1992]). Concentrat- ing on the main effects, the design of the experiments can be aimed at a minimum number of observations.

5.1.2 Experimental Design

Methods of variance analysis are helpful tools to evaluate effects of factors on the results of experiments afterwards. On the other hand, it may be advantageous to plan experiments in a comparative way (comparative experiments). Statistical experimental design is characterized by the three basic principles: Replication, Randomization and Blocking (block division, planned grouping). Latin square design is especially useful to separate nonrandom variations from random effects which interfere with the former. An example may be the identiˇcation of (slightly) different samples, e.g. sorts of wine, by various testers and at several days. To separate the day-to-day and/or tester-to-tester (laboratory-to-laboratory) variations from that of the wine sorts, an m m Latin square design may be used. In case of m = 3 all three wine samples (a, b, c) are tested be three testers at three days, e.g. in the way represented in Table 5.8:

Table 5.8. Latin square design for m =3

Tester 1 Tester 2 Tester 3 1st day abc 2nd day bca 3rd day cab

The results of the experiments are evaluated by means of three-way ANOVA in its simplest form, m = n = p and q = 1. The signiˇcance of the sample effect can principally be guaranteed also in the case that both testers and days have signiˇcant inuence (Sharaf et al. [1986]). In contrast to common statistical techniques, by modern experimental design inuencing factors are studied simultaneously (multifactorial design, MFD). The aim of MFD consists in an arrangement of factors in such a way that their inuences can be quantiˇed, compared and separated from random variations. Frequently the signal intensity of the analyte A is the target quantity, the inuences on which are described by Eq. (3.16a). Handling all the inuences (interferences and other factors) in the same way and holding xA at any constant value so that ˛0 = yA0 + SAAxA, Eq. (3.16a) can be written 5.1 Testing the Signiˇcance of Inuencing Factors 109

Table 5.9. Design matrix for three factors at two levels (+ and stand for +1 and 1)

Run (z0) z1 z2 z3 z1z2 z1z3 z2z3 z1z2z3 Target value

1 + +++ + + + + y1

2+++ + y2

3++ + + y3

4++ ++ y4

5+ ++ + y5

6+ + + + y6

7+++ + y7

8+ +++ y8

yA = ˛0 + ˛1x1 + ˛2x2 + ˛3x3 + ˛12x1x2 + ˛13x1x3 + ˛23x2x3 + ˛123x1x2x3 (5.6) in case of three inuence factors. From the various types of treatment and design, two-level factorial design is mostly applied. That means that the inuence factors are varied between a higher and a lower level, xmax and xmin. Using complete factorial design (CFD) the number of experiments is N =2m for m factors. In the case of m = 3 as given in Eq. (5.6), N =8 experiments have to be carried out. Expediently, factorial design is done on the basis of transformed factors zi, calculated from the xi by

xi x zi = 1 (5.7) 2 (xmax xmin)

1 xmaxxmin xminxmax where x = (xmax + xmin) so that zmax = = +1, zmin = = 2 xmaxxmin xmaxxmin 1, and z =0. With the transformation at Eq. (5.7), Eq. (5.6) becomes 2 y = a0(z0)+a1z1 + a2z2 + a3z3 + a12z1z2 + a13z1z3 + a23z2z3 + a123z1z2z3 (5.8)

The coefˇcients ai are estimated from the results of experiments carried out according to a design matrix such as Table 5.9 which shows a 23 plan matrix. The signiˇcance of the several factors are tested by comparing the coefˇcients with the experimental error, to be exact, by testing whether the conˇdence intervals ai include 0 or not. The experimental error can be estimated by repeated measurements of each experiment or { as it is done frequently in a more effective way { by replications at the centre of the plan (so-called \zero replications"), see Fig. 5.2. The coefˇcients are estimated according to 1 a = zT y (5.9) i N i

2 the target value yA is symbolized here y 110 5 Studying Inuences and Optimizing Analytical Procedures

Fig. 5.2. Geometrical representation of a complete two level factorial matrix (three inuence factors) with experiments in the centre point (0,0,0)

zT with i the corresponding transposed z vector and y the vector of target values obtained as the result of the runs. N is the number of experiments m (N =2, here N = 8). As an example, the coefˇcients ai in Eq. (5.8) corresponding to the design matrix at Eq. (5.9) are estimated by 1 a = (+y + y + y + y + y + y + y + y ) 0 8 1 2 3 4 5 6 7 8 1 a = (+y + y + y + y y y y y ) 1 8 1 2 3 4 5 6 7 8 1 a = (+y + y y y + y + y y y ) 2 8 1 2 3 4 5 6 7 8 1 a = (+y y + y y + y y + y y ) 3 8 1 2 3 4 5 6 7 8 1 a = (+y + y y y y y + y + y ) 12 8 1 2 3 4 5 6 7 8 . . 1 a = (+y y y + y y + y + y y ) 123 8 1 2 3 4 5 6 7 8 5.1 Testing the Signiˇcance of Inuencing Factors 111 according to the vector expression a = ZT y (5.10) with ZT being the transposed z matrix. Because the experimental expenditure increases strongly with the increasing number of inuence factors, fractional factorial design FFD (partial factorial design) is applied in such cases. It is not possible to evaluate all the interactions by FFDs but only the main effects. Plackett and Burman [1946] have developed a special fractional design which is widely applied in analytical optimization. By means of N runs up to m = N 1 variables (where some of them may be dummy variables which can help to estimate the experimental error) can be studied under the following prerequisites and rules:

The number of experiments (runs) must be a multiple of l2 (l is the number of levels), that is N =8; 12; 16; 24;:::in case of two-level experiments (l = 2). The ˇrst rows of the design matrixes are N =8: +++ + N =12: ++ ++++ N =16: ++++ + +++ etc. in case of two-level design. The following rows of the design matrix is generated by shifting the ˇrst row cyclically one place (N 2 times). The last row has minus in all factors. The procedure can be controlled as follows: each row contains m=2 times the higher level (+) and (m=2 1) times the lower (), the columns contain each m=2 times + and . Fractional factorial design is especially useful in case of a high number of inuence variables from which the insigniˇcant one have to be screened. An example of a Plackett Burman plan for l = 2 levels, m =7 inuence factors (including dummy variables) and, therefore, N = 8 runs is given in Table 5.10. The coefˇcients ai of the main effects of the model

y = a0z0 + a1z1 + a2z2 + a3z3 + a4z4 + a5z5 + a6z6 + a7z7 (5.11) are obtained by the vector equation a =(ZT Z)1ZT y (5.12) The coefˇcients characterize the effect of the belonging factor. The inuence is signiˇcant in the case that

jaij acrit = a = sa t1˛; (5.13) 112 5 Studying Inuences and Optimizing Analytical Procedures

Table 5.10. Plackett-Burman design matrix for N = 8 experiments and consequently m = 7 factors (including dummy variables) at two levels

Run z1 z2 z3 z4 z5 z6 z7 Target value

1 +++ + y1

2++ + + y2

3+ + ++ y3

4 + +++ y4

5++++ y5

6 +++ + y6

7 +++ + y7

8 y8

i.e., the inuence of zi (and therefore xi) is insigniˇcant if the conˇdence interval ai ˙ a includes zero. Multifactorial experiments with a low number of factors can also be evaluated by ANOVA (see Danzer et al. [2001], Sect. 5.1.1). If nonlinear effects are expected the variables must be varied at more than two levels. A screening plan comparable to the Plackett Burman design but on three levels is that of Box and Behnken [1960]. In case of special conditions, viz. internal correlations, interactions can be estimated in addition to the main effects by means of a 2m1 design. Multifactorial experiments are used in analytical chemistry for diverse applications, e.g., checking up signiˇcant inuences before optimization procedures, recognizing matrix effects, and testing the robustness of analytical procedures (Wegscheider [1996]).

5.2 Optimization of Analytical Procedures

Analytical procedures should always run under optimum conditions. That means that for Eq. (5.6), which is here used only with two factors, the coefˇcients have to be chosen in such a way that y becomes an optimum ! y = ˛0 + ˛1x1 + ˛2x2 = opt (5.14) In analytical chemistry the target quantity y which has to be optimized is frequently the signal intensity, absolute or relative (signal-to-noise ratio), but occasionally other parameters like yields of extractions or chemical reactions, too. The classical way to optimize inuences, e.g., in an optimization space as shown in Fig. 5.3a is to study the factors independently one after the other. In Fig. 5.3b,c it can be seen that an individual optimum will be found in this way. 5.2 Optimization of Analytical Procedures 113

Fig. 5.3. Contour plot of the response surface y = f (x1;x2); the optimum is situated at point (a); Response curves of y in dependence of x1 as result of a ˇrst run (b) and x2 as result of a second run (c)

However, the optima of x1 and x2 found in this way do not meet the global optimum of the response surface which is situated at x1 = 80 and x2 = 150. Because the global optimum is rarely found by such an obsolete proceeding, multivariate techniques of optimization should be applied. The most reliable technique to ˇnd the global optimum by means of common methods is the transition from the quasi-two-dimensional approach (Fig. 5.3b,c) to a complete two-dimensional one. It consists of a certain number of experiments as shown in Fig. 5.4. On the basis of the grid experiments a mathematical function y = f (x1;x2;x3;:::), called the response surface, is estimated that characterizes the response as a function of the factors. In case of only two factors the response surface can be visualized by plots like that in Fig. 5.5. Response surfaces are mostly described mathematically by polynomial approximations of 1st and 2nd degree. Grid search corresponds to a com- 114 5 Studying Inuences and Optimizing Analytical Procedures

Fig. 5.4. Grid experiments for estimating the response surface y = f (x1;x2)

Fig. 5.5. Response surface as a result of grid experiments according to Fig. 5.4 plete factorial design. When the resolution of the variables is high enough, each optimum { the global one and all the local maxima and minima { can be found. But the high number of experiments imposes limits in this regard how it is generally in response surface technique. The number of experiments can considerably be decreased by itera- tive optimization methods which starts at an area that can be selected by experience, supposition or randomly. This start area is moved step by 5.2 Optimization of Analytical Procedures 115

Fig. 5.6. Schematic representation of the Box-Wilson optimization with step width h step in direction to the optimum following the gradient of the function 3 y = f (x1;x2;x3;:::), i.e. the steepest ascent . The well-known Box-Wilson optimization method (Box and Wilson [1951]; Box [1954, 1957]; Box and Draper [1969]) is based on a linear model (Fig. 5.6). For a selected start hyperplane, in the given case an area A0(x1;x2), described by a polynomial of ˇrst order, with the starting point y0, the gradient grad[y0] is estimated. Then one moves to the next area in direction of the steepest ascent (the gradient) by a step width of h,in general

yi+1 = yi + h grad[yi] (5.15) Near the optimum both the step width and the model of the hyperplane are changed, the latter mostly from a ˇrst order model to a second order model. The vicinity of the optimum can be recognized by the coefˇcients a1;a2;::: of Eq. (5.14) which approximate to zero or change their sign, respectively. For the second order model mostly a Box-Behnken design is used. Because this proceeding is relatively expensive, an effective semi- quantitative method is widely used in optimization, the sequential simplex optimization. Simplex optimization is done without estimation of gradients and setting step widths. Instead of this, the progress of the optimization

3 steepest descent in case of minima 116 5 Studying Inuences and Optimizing Analytical Procedures Table 5.11. Basic simplex operations

Operation Movement Condition

Reexion 1 4 y1

Expansion 4 5 y4 >y2 y3

3 2

Contraction 6 4 y4 y2 y3

3 2

Strong contraction 7 4 y4

3 procedure results directly from the quality of the preceded experimental values. A simplex is a geometric ˇgure formed by p +1 points in a p-dimensional space. In the two-dimensional case y = f (x1;x2) the simplex is a triangle, in the three-dimensional case a tetrahedron etc. With regard to its form, the simplex may be regular, rectangular, or irregular. The simplex optimization starts with a set of p + 1 parameters (here p + 1 = 3). The movement of the simplex takes place according to the rules given in Table 5.11. As an example, in Fig. 5.7 a simplex optimization is shown in a sim- pliˇed way, i.e., only by reexions and with simplexes of invariable size. The approach to the optimum is indicated by rotation or oscillation of the simplex. Then contractions should be included into the operations. The optimum found by sequential proceeding, both by Box-Wilson and simplex technique, is that local optimum situated nearest the starting point. It must not inevitably be identical with the global optimum. Therefore, it may be useful to repeat the optimization procedure one or several times.

5.3 Global Optimization by Natural Design

Some natural processes and principles have stimulated researchers to develop algorithms that imitate concepts of nature and are, therefore, summarized under the name natural computation (Kateman [1990]; Lucasius [1994]). The most prominent methods are: 5.3 Global Optimization by Natural Design 117

Fig. 5.7. Simplex optimization within a response surface of two factors x1 and x2 with simplexes of invariable size

Thermal computation which comprises algorithms inspired by multi- particle systems like Brownian motion (Brownian search, Monte Carlo search) on the one hand and simulated annealing which is inspired by Boltzmanns statistics on the other hand. Evolutionary computation which is learned by watching population dynamics; the most important programming are genetic algorithms which are inspired by the evolutionary processes of mutation, recombination, and natural selection in biology. Connectionist computation which is inspired by multi-cellular systems like artiˇcial neural networks that mimic the way of working of the human brain. Simulated annealing is a category of global optimization procedures the name of which is derived from the statistical mechanics simulating the atomic equilibrium according to the Boltzmann statistics. By means of simulated annealing it is searched for the most probable conˇguration of parameters on the basis of simulating the evolution of a substance to thermal equilibrium (Kirkpatrick et al. [1983]; Frank and Todeschini [1994]). The distribution of conˇgurations x can be described by the Boltzmann distribution exp (C(x)=c) P (x)= (5.16) exp (C(xw )=c) w 118 5 Studying Inuences and Optimizing Analytical Procedures where C(x) is the function to be optimized, xw are other conˇgurations and c is a control parameter. From the initial conˇguration x another conˇgu- ration xr in the neighborhood of x is generated by modifying one randomly selected variable. The new conˇguration is accepted when the difference C(xr ; x) 0, otherwise the probability

P = exp (C(xr ;x)=c) (5.17) is compared with a random number generated from a uniform distribution [0; 1]. If P is larger than that random number, the new conˇguration is also accepted, otherwise it is declined. The iteration is continued until convergence is reached. Afterwards the optimization runs are continued with lowered control parameter c. More detailed information on simulated annealing can be found in van Laarhoven and Aarts [1987]; Kalivas [1992]. Genetic Algorithms (GA) are the most important global optimization techniques. GA base on mimicking the evolution process by variation of populations according to the Darwin rules [Darwin 1859] such as selection, reproduction, crossover (recombination), and mutation. Genetic algorithms have been pioneered by Holland [1975], detailed representations can be found in Goldberg [1989]; Davis [1991], Rechenberg [1973]. The initial data are binary coded in form of a bit sequence (bit string4). Start values of the variables x1;x2,...,xm could be, e.g., 011001011; 100101100, :::;010010101. This initial population is undertaken an evolution process such as schematically represented in Fig. 5.8. In the course of each run corresponding to Fig. 5.8 the ˇtness of the members of the population is tested by means of an objective criterion (e.g., maximum correlation of a regression model or minimum random deviation of a response surface) that is compared with a break-off criterion ˇxed in advance. According to their ˇtness, the members from the present population (generation) are selected and reproduced by doubling. On the other hand, less ˇt members are omitted from the population. In the recombination step, parts of the bit-string are exchanged, namely by single-, two- or three-point-, uniform-, or circular crossover. In this way (by \mating"), from two parent bit strings two offspring are generated. Finally, by mutation only a small number of genes from the whole population is changed by ipping to the opposite value (0 ! 1 and 1 ! 0, respectively). For the selection and reproduction step the idea of \elitism" plays a role in so far as individuals of high quality should not become extinct. On the other hand, a larger number of elitists produces untimely a homogenisation of the population. The advantage of Genetic Algorithms, in contrast to the traditional optimization methods, is the fact that a large number of variables can be included into the process. Also in the presence of local optima, GA can ˇnd rapidly the global optimum.

4 Also called \chromosome"; a bit in this chromosome is then called \gen". 5.3 Global Optimization by Natural Design 119

Fig. 5.8. Flow chart of a Genetic Algorithm

In many practical problems, interactions between the variables appear so that the absolute global optimum can be found heavily. As an example, wavelength selection in NIR determination of blood glucose (see Sect. 6.2.6) is considered. The aim of the selection is to ˇnd such combinations of wavelengths with which calibration models are obtained their prediction quality is as near at the global optimum as possible (Danzer et al. [2001], p 174). The number of combinations C for the selection of k wavelengths from n channels of the spectrometer is given by n n! C = = (5.18) k k!(n k)!

So, for the selection of 15 wavelengths out of 86 the number of 2:17841016 combinations can be generated. Because of the existence of background, noise and strong multicollinearities in the NIR spectra, a large number of quasi-optimum solutions are available between which cannot be differentiated signiˇcantly. In Fig. 5.9 the wavelength selection 15 out of 86 is shown, carried out to improve the quality of the calibration model. Whereas the 2 2 2 cross validated PRESS value s(c)res (see Eq. 6.105) is 295.6 mmol /L when 2 2 86 equidistant wavelengths are used, PRESSCV improves to 147.0 mmol /L for 15 GA-selected wavelengths. 120 References

Fig. 5.9. Wavelength selection by Genetic Algorithm: a 15 optimum wavelength selected from 86 of the full spectrum; b relative ˇtness (rf/104) of the run in dependence of the number of generations, above: ˇtness of the best solution, middle: mean, below: ˇtness of the worst solution (Fischbacher et al. [1994/96;1995])

In connection with the NIR determination of blood glucose, GA has been used also to select spectra according to the quality of recordings (Fischbacher et al. [1994/96;1995]). Optimization problems in general and, in particular in analytical chemistry, have been reported by Goldberg [1989]; Davis [1991]; Lucasius et al. [1991]; Lucasius and Kateman [1993]; Wienke et al. [1992, 1993], and others. In analytical chemistry, Artiˇcial Neural Networks (ANN) are mostly used for calibration, see Sect. 6.5, and classiˇcation problems. On the other hand, feedback networks are usefully to apply for optimization problems, especially nets of Hopfield type (Hopfield [1982]; Lee and Sheu [1990]).

References

Ahrens H (1967) Varianzanalyse. Akademie-Verlag, Berlin Box GPE (1954) The exploration and exploitation of response surfaces: some general considerations and examples. Biometrics 10:16 Box GEP (1957) Evolutionary operation: a method for increasing industrial productivity. Appl Statist 6:81 Box GEP, Behnken DW (1960) Some new three level designs for the study of quantitative variables. Technometrics 2:445 Box GEP, Draper NR (1969) Evolutionary operation. Wiley, New York Box GPE, Wilson KB (1951) On the experimental attainment of optimum conditions.JR Statist Soc B 13:1 Danzer K, Marx G (1979) Application of the two-dimensional variance analysis for the investigation of homogeneity of solids. Anal Chim Acta 110:145 Danzer K, Hobert H, Fischbacher C, Jagemann K-U (2001) Chemometrik { Grundlagen und Anwendungen. Springer, Berlin Heidelberg New York Darwin C (1859) On the origin of species by means of natural selection, or the preservation of favoured races in the struggle for life. John Murray, London References 121

Davis L (ed) (1991) Handbook of genetic algorithms. Part I. A genetic algorithms tutorial. Van Nostrand Reinhold, New York Dixon W, Massey FJ (1983) Introduction to statistical analysis. McGraw-Hill, New York Dunn OJ, Clark VA (1974) Applied statistics - analysis of variance and regression. Wiley, New York Eisenhart C (1947) The assumptions underlying the analysis of variance. Biometrics 3:1 Fischbacher C, Jagemann K-U, Danzer K, Muller UA, Mertes B, Papenkorth L, Schuler J (1994/96) Unpublished results Fischbacher C, Jagemann K-U, Danzer K, Muller UA, Mertes B (1995) Non-invasive blood glucose monitoring by chemometrical evaluation of NIR-spectra. 24th EAS (Eastern Analytical Symposium), Somerset, NJ, November 12-16 Fisher RA (1925) Theory of statistical estimation. Proc Cambridge Phil Soc 22:700 Fisher RA (1935) The design of experiments. Oliver & Boyd, Edinburgh (7th edn 1960; 9th edn 1971, Hafner Press, New York) Frank IE, Todeschini R (1994) The data analysis handbook. Elsevier, Amsterdam Goldberg DE (1989) Genetic algorithms in search, optimization and machine learning. Addison-Wesley, New York Graf U, Henning H-U, Stange K, Wilrich P-Th (1987) Formeln und Tabellen der angewandten mathematischen Statistik. Springer, Berlin Heidelberg New York (3. Au) Hald A (1960) Statistical tables and formulas. Wiley, New York Holland JH (1975) Adaption in natural and artiˇcial systems. University of Michigan Press, Ann Arbor, MI. Revised ed (1992) MIT Press, Cambridge, MA Hopˇeld JJ (1982) Neural networks and physical systems with emergent collective computational abilities. Proc Nat Acad Sci USA 79:2554 Kalivas JH (1992) Optimization using variations of simulated annealing. Chemom Intell Lab Syst 15:1 Kateman G (1990) Evolutions in chemometrics. Analyst 115:487 Kirkpatrick S, Gelatt CD, Vecchi MP (1983) Optimization by simulated annealing. Science 220:671 Lee BW, Sheu BJ (1990) Combinatorial optimization using competitive Hopˇeld neural network. Proc Internat Joint Conf Neural Networks II: 627, Washington, DC Lucasius, CB (1994) Evolutionary computation: a distinctive form of natural computation with chemometric potential. Chapter 9 in: Buydens LM , Melssen WJ: Chemometrics. Exploring and exploiting chemical information. Katholieke Universiteit Nijmegen Lucasius CB, Kateman G (1993) Understanding and using genetic algorithms. Part 1. Concepts, properties and context. Chemom Intell Lab Syst 19:1 Lucasius CB, Blommers MJJ, Buydens LMC, Kateman G (1991) A genetic algotithm for conformational analysis of DNA. In: Davis L (ed) Handbook of genetic algorithms.Van Nostrand Reinhold, New York Mandel J (1961) Non-additivity in two-way analysis of variance. J Am Statist Assoc 56:878 Neave HR (1981) Elementary statistical tables. Allen Unwin, London 122 References

Plackett RL, Burman JP (1946) The design of optimum multifactorial experiments. Biometrika 33:305 Rechenberg I (1973) Evolutionsstrategie. Optimierung technischer Systeme nach Prinzip- ien der biologischen Evolution. Frommann-Holzboog, Stuttgart Sachs L (1992) Angewandte Statistik, 7th edn. Springer, Berlin Heidelberg New York Scheffe H (1961) The analysis of variance. Wiley, New York Sharaf MA, Illman DL, Kowalski BR (1986) Chemometrics. Wiley, New York van Laarhoven PJM, Aarts EHL (1987) Simulated annealing: theory and applications. Reidel, Dordrecht Wegscheider W (1996) Validation of analytical methods. In: H. Gunzler (ed) Accreditation and quality assurance in analytical chemistry. Springer, Berlin Heidelberg New York Wienke D, Lucasius C, Kateman G (1992) Multicriteria target vector optimization of analytical procedures using a genetic algorithm. Part I. Theory, numerical simulations and application to atomic emission spectroscopy. Anal Chim Acta 265:211 Wienke D, Lucasius C, Ehrlich M, Kateman G (1993) Multicriteria target vector optimization of analytical procedures using a genetic algorithm. Part II. Polyoptimization of the photometric calibration graph of dry glucose sensors for quantitative clinical analysis. Anal Chim Acta 271:253 6 Calibration in Analytical Chemistry

In measurement sciences, calibration is an operation that establish a relationship between an output quantity, qout, with an input quantity, qin, for a measuring system under speciˇed conditions (qin ! qout). The result of calibration is a model that may have the form of a conversion factor, a mathematical equation, or a graph. By means of this model, then it is possible to estimate qin-values from measured qout-values (qout ! qin)ascan be seen in an abstracted form in Fig. 6.1.

Fig. 6.1. Abstract representation of the calibration procedure by conversion of qin into qout and the reverse evaluation procedure

From the information-theoretical point of view, calibration corresponds to the coding of the input quantity into the output quantity and, vice versa, the evaluation process corresponds to decoding of output data. From the mathematical viewpoint, qin is the independent quantity in the calibration step and qout the dependent one. In the evaluation step, the situation is reverse: qout is the independent, and qin the dependent quantity. From the statistical standpoint, qout is a random variable both in calibration and evaluation whereas qin is a ˇxed variable in the calibration step and a random variable in the evaluation step. This rather complicated situation has some consequences on which will be returned in Sect. 6.1.2. 124 6 Calibration in Analytical Chemistry

Table 6.1. Input quantities, qin, and output quantities, qout, in analytical calibration

qin qout Analytical technique Calibration function xyQuantitative analysis y = f (x) (see Sect. 6.2) QzIdentiˇcation, qualitative analysis z = f (Q) (see Sect. 2.4) x(Q) y (z) Quantitative multicomponent Y = f (X) (see Sect. 6.4) analysis I, S y (z) Identity, structure analysis Comparison with reference signal functions (see Sect. 2.4)

6.1 General Fundamentals of Calibration

In analytical chemistry, calibration represents a set of operations that con- nects quantities in the sample domain with quantities in the signal domain (see Sect. 2.3, Fig. 2.12). In Table 6.1 the real analytical quantities and properties behind the abstract input and output quantities are listed. Calibration in analytical chemistry relates mostly to quantitative analysis of selected species and, therefore, to calibration functions of the kind y = f (x).

6.1.1 Fundamental and Experimental Calibration

Depending on the type of relationships between the measured quantity and the measurand (analytical quantity) it can be distinguished (Danzer and Currie [1998]) between calibrations based on absolute measurements (one calibration is valid for all1 on the basis of the simple proportion y = b x, where the sensitivity factor b is a fundamental quantity; see Sect. 2.4; Hula- nicki [1995]; IUPAC Orange Book [1997, 2000]), deˇnitive measurements (b is given either by a fundamental quantity complemented by an empirical factor or a well-known empirical (transferable) constant like molar absorption coefˇcient and Nernst factor), and experimental calibration. In the simplest case, experimental calibration can be carried out by direct reference measurements where the sensitivity factor b is given by the relation of measured value to concentration of a reference material (RM), b = yRM=xRM. Direct reference calibration is frequently used in NAA and X-ray analytical techniques (XRF, EPMA, TXRF). On the other hand, indirect reference measurements which result in an empirical calibration function, frequently based on a linear model, y = a + b x + ey ; see Sect. 2.4, Eq. (2.22), where the intercept a corresponds to

1 Apart from the fact that both the fundamental constants b and the conditions under which they are valid (e.g. under which a reaction proceeds quantitatively) sometime were determined and have to be updated from time to time (e.g., on the basis of IUPAC documents). 6.1 General Fundamentals of Calibration 125 the experimental blank and the slope b to the experimental sensitivity; ey is the error of the y -measurement. In analytical practice, some methods using deˇnitive measurements, in principle, are also calibrated by indirect reference measurements using least squares estimating to provide reliable estimates of b (spectrophotometry, potentiometry, ISE, polarography). Because calibration is the prerequisite of reliable evaluations and, therefore, of analytical results which are both accurate and precise, calibration itself has to be carried out in a very reliable way. For this reason, the following experimental and fundamental conditions have to be realized: (i) The standards used for the calibration measurements should be both reliable and traceable. Traceability is the property of the result of a measurement or the value of a standard whereby it can be related to stated references, usually national or international standards, through an unbroken chain of comparisons all having stated uncertainties (ISO [1993]; Hasselbarth [1995]). The connection to the International Sys- tem of Units (SI, ISO [1993]) is given in particular by the amount of substance (mole) and mass (kg) and is realized in detail by certiˇed reference materials (CRMs). If laboratory standards are used for calibration then these have to be validated by CRMs. (ii) The measurement strategy of experimental calibration have to be ˇxed. Speciˇcally: { The independence of the diverse calibration samples have to be guaranteed, that means that each sample must be prepared separately and not by down-diluting of a master sample. { The number of calibration points, p , their distance and measure at the concentration scale, the number of replicate measurements, ni, in each point carried out with independent calibration samples (and not by repetition measurements on one and the same sample), and their distribution at the calibration points (equally distributed, n1 = n2 = :::= np , or in large numbers at the ends of the calibration range, n1 >n2 = n3 = :::= np 1

Fig. 6.2. Real representation of the calibration procedure in analytical chemistry

As a consequence, the relationship between y on the one hand and xcalibr and xsample on the other hand has to be represented in a three-dimensional way (Danzer [1995]).

6.1.2 The General Three-Dimensional Calibration Model

The three-dimensional model of calibration and the spatial relationships of calibration, evaluation, and recovery is given in Fig. 6.3. It should be noted that the spatial model contains additionally to the calibration and evaluation functions the recovery function, too. These three functions represent the projection of the spatial C-E-R function onto the corresponding planes, as shown in Fig. 6.3. From the different character of xstandard and xsample (Figs. 6.2 and 6.3) it can be seen that the relationship between the calibration function

y = f (x)=a + b x + ey (6.1) and the evaluation function y a x = f 1(y )= + e (6.2) b x must not be necessarily reversible. This fact has to be checked by validation. Commonly the equivalence is tested by recovery experiments to rule out systematic deviations (see Sect. 4.1.1). The conˇrmation that xtest = xtrue, i.e. xstandard = xsample in the calibration procedure, is an important step to ensure the traceability of analytical results on the basis of the given calibration. Only in the case when the mutual equivalence of Eqs. (6.1) and (6.2) in the three-dimensional model have been proved, it can be used { as usual { in a two-dimensional way. Pictorially this may be imagined by folding up the planes of the calibration function and of the evaluation function (B and C in Fig. 6.3) to a single plane containing a common calibration-evaluation function as it is usual in analytical chemistry. Consequences of various types of systematic deviations between xtest and xtrue, and therefore, xstandard and xsample are discussed in Danzer [1995]. The different statistical character of the three variables becomes most clear in the different uncertainties of the calibration and evaluation lines. Notwithstanding the fundamental differences between xstandard and xsample, the calculation of the calibration coefˇcients is carried out by regression calculus. 6.1 General Fundamentals of Calibration 127 A B

y y xsample

xstandard xstandard

C D

xsample xsample

xstandard y

Fig. 6.3. Three-dimensional model of calibration, analytical evaluation and recovery: spatial model (A); the three relevant planes are given separately in (B) as the calibration function with conˇdence interval, in (C) as the recovery function with conˇdence interval, and in (C) as the evaluation function with prediction interval (D)

6.1.3 Regression and Calibration Having two random variables, x and y , e.g. values measured independently from each other and each of them being normally distributed, then the following questions may be of interest: (i) Does a relation exist between y and x and how strong is it? (ii) How the relation can functionally be described, i.e. how can y be estimated from x and vice versa? Whereas the ˇrst question is answered by correlation analysis, the second one is subject of regression analysis. Correlation analysis investigates stochastic relationships between random variables on the basis of samples. The interdependence of two variables x 128 6 Calibration in Analytical Chemistry and y is characterized by the correlation coefˇcient rxy (Pearson's correlation coefˇcient)2. For the calculation of measures in correlation and regression analysis the following sums are of relevance: n n 2 2 Sxx = (xi x) ;Syy = (yi y ) ; i=1 i=1 (6.30) n Sxy = (xi x)(yi y ) i=1 With this the correlation coefˇcient of x and y is calculated by & Sxy %sxy sxy rxy = = = (6.3) Sxx Syy 2 2 sx sy sx sy As the last term in Eq. (6.3) shows, the correlation coefˇcient corresponds to the covariance of x and y , cov(x;y)=sxy, divided by the standard deviations sx and sy . Concrete values of correlation coefˇcients indicate the following situations:

rxy = +1, rxy = 1 perfect (positive and reverse, respectively) interdependence of x and y

0 < jrxy j < 1 stochastic interdependence

rxy = 0 missing dependence.

The pairwise correlation of more than two variables x1;x2;:::;xm is characterized by the correlation matrix R ⎛ ⎞

1 rx1x2 rx1xm ⎜ r 1 r ⎟ ⎜ x2x1 x2xm ⎟ R = ⎝ . . . ⎠ (6.4) . . .

rxm x1 rxm x2 1 In multivariate data analysis frequently the covariance matrix S is used ⎛ ⎞ var(x1) cov(x1;x2) cov(x1;xm) ⎜ ⎟ ⎜ cov(x2;x1) var(x2) cov(x2;xm) ⎟ S = ⎝ . . . ⎠ . . . cov(xm;x1) cov(xm;x2) var(xm) ⎛ ⎞ (6.5) 2 sx1 sx1x2 sx1xm 2 ⎜ sx x s sx x ⎟ ⎜ 2 1 x2 2 m ⎟ = ⎝ . . . ⎠ . . . 2 sx1x2 sx1x2 sxm

2 The correlation coefˇcient rxy of a sample is an estimate of the correlation coefˇcient xy of the population. 6.1 General Fundamentals of Calibration 129 In contrast to correlation matrix the covariance matrix is scale-dependent. In case of autoscaled variables the covariance matrix equals the correlation matrix. In analytical calibration, there exists { strictly speaking { no correlation problem for the following two reasons: (1) The interdependence of the measured values, y , and the analytical values, x, is well-known a priori { mostly by natural laws { and is, therefore, not subject of veriˇcation as a rule

(2) The analytical values of the calibration standards, xstandard, are no random variables but ˇxed one and carefully selected. Two consequences result from this fact:

The correlation coefˇcient, which is a characteristic for the relationship between random variables, is not meaningful in calibration (Currie [1995]; Danzer and Currie [1998]) and should, therefore, not be used to characterize the quality of calibration (instead of rxy the residual standard deviation sy:x should be applied; see Eq. (6.19)). From the diverse possible regression calculi, a certain algorithm has to be selected, namely that of the regression of y onto x; see Eq. (6.6). The general pre-conditions for the estimation of regression parameters (regression coefˇcients a and b as well as the uncertainties of the model and the relevant estimates) are the following: (i) Real replication measurements must be carried out (runs with the same treatment), not only repeated measurements at the same samples (ii) Normal distribution of the x- and y -values (iii) Homoscedasticity, i.e. homogeneity of variances in the diverse calibration points Depending on whether x can be considered to be free or almost free of 3 errors (sx sy =b) or vice versa (sy =b sx ), a regression of y onto x

y^ = ax + bx x (6.6) by minimizing the y -deviations according to the Gaussian least squares criterion or a regression of x onto y

x^ = ay + by y (6.7) by minimizing the xdeviations have to be carried out. The situation is illustrated in Fig. 6.4 from which can be seen that the re- version of the dependent and independent variables gives different estimates y^ and x^. The regression coefˇcients in Eq. (6.6), bx and ax , for the estimation of y from x are estimated by the following expressions:

3 This model usually does not have any relevance in analytical calibration. 130 6 Calibration in Analytical Chemistry

Fig. 6.4. Different regression models as a result of the minimization of the x- and the y -deviations, respectively

Sxy b = (6.8) x S xx y b x a = x : (6.9) x n

Because of the irrelevant condition sy =b sx ), the reverse model, viz. the estimation of x from y , according to Eq. (6.7) and therefore the co- efˇcients by and ay , are not of direct relevance in analytical calibration. Notwithstanding, their estimates will be given here for completion and as auxiliary quantities for further calculations: Sxy x by y by = ;ay = : (6.10) Syy n All the special cases that may be appear in analytical calibration in dependence from the speciˇc conditions as well as the uncertainties of the calibration measures will be given in the following.

6.2 Single Component Calibration

6.2.1 Linear Calibration Model

On the condition that the errors of the measurement have a zero mean and are uncorrelated, the linear function (Eq. 6.1) can be ˇtted to the measured 6.2 Single Component Calibration 131 values by means of the Gaussian least squares estimation (GLS, or ordinary least square estimation, OLS, respectively; see Danzer and Currie [1998]). With the fundamental relations

Model: yi = a + bxi + eyi (6.11a) ^ Estimate: y^i = a^ + bxi (6.11b)

^ Residual: eyi = yi y^i = yi a^ bxi (6.11c) the general least squares criterion expressed by the sum of squares of residuals, ssr(y ), reads (Danzer [1990]) n 2 n 2 y y^ ey ssr(y )= i i = i (6.12) si si i=1 i=1 where si is the estimated standard deviation at the given point i and n the total number of calibration measurements which are equal to the sum of measurements nj in each calibration point p p n = nj (6.13a) j=1 which is simpliˇed to

n = p nj (6.13b) if an equal number nj of repetitions in each calibration point is carried out. The sum of squares of residuals has to be minimized according to the general least squares (LS) criterion n 2 ey ! i = min (6.14) si i=1 Depending on the fulˇlment of the conditions mentioned above, namely (i) to (iii) and following, the least squares criterion has to be modiˇed as follows: 1. The errors are only or essentially in the measured values y as the dependent variable (bsx sy ) and in addition, the errors sy are constant in the several calibration points (Homoscedasticity):

2 =˛ 2 =˛ =˛ 2 =˛ 2 sy1 sy2 syp sy (6.15)

˛ where = means equality for a given risk of error ˛. Only in this homoscedastic case and if errors in x can be neglected, the LS criterion is reduced to n ! e2 = min (6.16) yi i=1 132 6 Calibration in Analytical Chemistry and the ordinary least squares estimation of the calibration model according to Eqs. (6.6), (6.8) and (6.9) can be applied. The evaluation of analyses carried out on the basis of the inverted calibration function (Eq. 6.6) y^ a^ x^ = (6.17) b^ and not according to Eq. (6.7) that holds for other statistical assumptions. On the other hand, Centner et al. [1998] found that predictions on the basis of inverse calibration (Eq. 6.7) may be more reliable, especially in the case of less precise measurements.

2. In case that the measuring errors sy vary and therefore heteroscedasticity must be assumed, the original LS criterion (Eq. 6.14) must be applied and the model of weighted least squares (WLS) results from this criterion as will be shown in Sect. 6.2.3. 3. In the most general case, if both variables are subject to error and, therefore, neither the condition sx sy =b nor sy =b sx is fulˇlled, the following total error results: s2 = s2 + b2 s2 : (6.18) i yi xi In this case, in which there are errors in both variables, y and x, the 2 resulting error ex+y has to be minimized (see Fig. 6.5) and orthogonal least squares ˇtting must be carried out as will be shown in Sect. 6.2.4. 4. Sometimes the basic conditions for the use of LS are not fulˇlled, either by measurement values being not-normally distributed or by strongly

Fig. 6.5. Regression model of errors in both variables in comparison to the model given in Fig. 6.4 6.2 Single Component Calibration 133 deviating calibration points (leverage points, see Rousseeuw and Leroy [1987]). In such cases, methods of robust calibration may be used; see Sect. 6.2.6. 5. In addition to statistical peculiarities, special features may also result from certain properties of samples and standards which make it necessary to apply special calibration techniques. In cases when matrix effects appear and matrix-matched calibration standards are not available, the standard addition method (SAM, see Sect. 6.2.6) can be used. Measurements in living things, e.g. patients, is occasionally done by sampling in a non-invasive way. Because no \standard patients" are available which can be selected for calibration, another principle of calibration must be applied, e.g., by a reference method that measures parallel to the actual measuring method. So, the calibration consists in a comparison of the measured results with that of the reference method which are considered to be true (that have been validated beforehand). An example of a reference calibration is represented by non-invasive blood glucose determination by means of NIR spectroscopy (Muller et al. [1997]; Fischbacher et al. [1997]; Danzer et al. [1998], see Sect. 6.2.6).

The several variants deriving from the items 1 to 4 are represented in the ow sheet given in Fig. 6.6. Common calibration by Gaussian least squares estimation (OLS) can only be applied if the measured values are independent and normal-distributed, free from outliers and leverage points and are characterized by homoscedastic errors. Additionally, the error of the values in the analytical quantity x (measurand) must be negligible compared with the errors of the measured values y . From the chemical point of view, in cases where matrix effects appear and no suitable certiˇed reference materials are available, the calibration may be performed in the sample matrix itself by means of standard addition. In analytical practice, linear calibration by ordinary least squares is mostly used. Therefore, the estimates are summarized before the uncertainties of the estimates will be given:

OLS estimates: y^ = ax + bx x (6.6)

y^ a Evaluation estimate: x^ = x (6.17) bx

Sxy Slope (sensitivity): bx = (6.8) Sxx y b x Intercept (\blank"): a = x (6.9) x n 134 6 Calibration in Analytical Chemistry

Fig. 6.6. Flow sheet of several calibration models in dependence of the fulˇlment of certain statistical and chemical conditions

6.2.2 Errors in Linear Calibration

Fundamentally, the uncertainties of measured values y^ estimated by calibration, e.g. according to Eq. (6.6), on the one hand and of analytical results x^ (analyte contents, concentrations) estimated by means of a calibration function, e.g. according to Eq. (6.17), on the other hand differ from one another as can be seen from Fig. 6.3B,C, and Fig. 6.7. Whereas the uncertainty of y values in calibration is characterized by the conˇdence interval cnf(y^), the uncertainty of estimated x values is characterized by the prediction interval prd(x^). 6.2 Single Component Calibration 135

= ˆ + ∆ ˆ prdup (yˆ) y ( y) prd = ˆ + ∆ ˆ y cnfup (yˆ) y ( y)cnf yˆ = aˆ + bˆ x = ˆ − ∆ ˆ cnflow( yˆ) y ( y)cnf = ˆ − ∆ ˆ prdlow (yˆ) y ( y) prd

+ ∆ aˆ ( aˆ) prd + ∆ aˆ ( aˆ)cnf

aˆ

Fig. 6.7. Calibration straight line with relevant upper and lower conˇdence and prediction bands

The uncertainty of OLS calibration is characterized by the following special standard deviations and uncertainty intervals: (1) Residual standard deviation n n 2 ^ 2 (yi y^i) (yi a^x bx xi) s = i=1 = i=1 (6.19) y:x n 2 n 2

In calibration, the number of degrees of freedom depends on the number of parameters estimated by the given model. In case of the two-parametric model (Eq. 6.6) = n 2, in case of linear calibration through the coordinate origin (a =0) = n 1, and in case of a three-parametric nonlinear calibration model (y = a + bx + cx2) = n 3.

(2) Estimated standard deviation (ESD) of the estimated intercept (blank) a^ " 1 x2 sa^ = sy:x + (6.20) n Sxx

with n being the total number of calibration measurements, x the mean of all the x values of the calibration experiment, and Sxx the sum of the squared x deviations. 136 6 Calibration in Analytical Chemistry (3) ESD of the estimated slope b^

sy:x sb^ = p (6.21) Sxx

(4) ESD of an estimated mean y^ at position xi " 2 1 (xi x) sy^ = sy:x + (6.22) n Sxx

(5) ESD of a predicted single value y^prd at position xi " 2 1 (xi x) sy^prd = sy:x 1+ + (6.23) n Sxx

^ (6) ESD of a predicted mean y prd from N repetitions in position xi " 2 1 1 (xi x) sy^ = sy:x + + (6.24) prd N n Sxx

(7) ESD of a predicted mean x^prd from N repetitions for a measured value yi " 2 sy:x 1 1 (yi y ) s^ = + + (6.25) xprd 2 b N n b Sxx

(8) The conˇdence band, CB, of the entire calibration straight line as shown in Fig. 6.7 is given by &

CB = y^ ˙ sy^ 2F1˛;1=2;2=n2 (6.26a)

and the prediction band, PB &

PB= y^prd ˙ sy^prd 2F1˛;1=2;2=n2 (6.26b)

The following uncertainty intervals resulting from Eqs. (6.19) to (6.25) are of practical interest: (9) Conˇdence interval of the intercept (blank) a^

cnf(a^)=a^ ˙ sa^ t1˛;=n2 (6.27) 6.2 Single Component Calibration 137 (10) Prediction interval prd(a) of a single a-value by Eq. (6.28) and that of an average a from N repetition measurements according to Eq. (6.29) " 1 x2 prd(a)=a ˙ sy:x t1˛;=n2 1+ + (6.28) n Sxx " 1 1 x2 prd(a)=a ˙ sy:x t1˛;=n2 + + (6.29) N n Sxx The latter is important with regard to the estimation of the detection limit from blanks ^ (11) Conˇdence interval of an estimated mean y i at position xi " 2 ^ ^ 1 (xi x) cnf(y i)=y i ˙ sy:x t1˛;=n2 + (6.30) n Sxx

(12) Prediction interval of a single value y^prd at position xi " 2 1 (xi x) prd(y^i)=y^i ˙ sy:x t1˛;=n2 1+ + (6.31) n Sxx

^ (13) Prediction interval of a mean y prd from N repetitions at position xi " 2 ^ ^ 1 1 (xi x) prd(y i)=y i ˙ sy:x t1˛;=n2 + + (6.32) N n Sxx

(14) Prediction interval of a mean x^ prd from N repetitions for a measured value yi " s 1 1 (y y )2 ^ ^ ˙ y:x i prd(xi)=xi t1˛;=n2 + + 2 (6.33) b N n b Sxx

6.2.3 Weighted Linear Least Squares Estimation (WLS)

In cases in that homoscedasticity according to Eq. (6.15) is not given, the estimated standard deviation is frequently a function of the measured quantity, sy = f (y ). In this case the calibration system is heteroscedastic and weighted least squares ˇtting has to be applied (Garden et al. [1980]; Draper and Smith [1981]). In general, least squares ˇtting starts from Eq. (6.14) and only in the case of homoscedasticity Eq. (6.16) can be taken as the basis. 138 6 Calibration in Analytical Chemistry From Eq. (6.14) it suggests itself that squared variance weighting w 1=s2 yi yi should be applied. In analytical practice, frequently relative weights 1 2 sy w = i (6.34) yi p 1 1 p s2 i=1 yi are used for the p calibration points in which each nj repetitions are made, see Eq. (3.13a,b). The minimizing criterion (Eq. 6.14) now becomes n ! w e2 = min (6.35) yi yi i=1 and the calibration coefˇcients are calculated analogous to Eqs. (6.8) and (6.9) by means of the weights (w = w ) yi i n wixiyi wixi wiyi bx = ! (6.36) w 2 2 n wixi wixi w y b w x a = i i xw i i (6.37) xw n As a rule the calibration coefˇcients do not alter signiˇcantly when WLS is applied instead of OLS. On the other hand, the uncertainty in the lower range of calibration is reduced in a remarkable way. The estimate of the residual standard deviation is $ w (y y^ )2 s = i i i (6.38) (y:x)w n 2 Other quantities characterizing uncertainties can be estimated in analogy to Eqs. (6.20){(6.33). Today, software packages for regression analysis usually allow one to enter an estimate of the functional dependence sy = f (y ) and to carry out a suitable weighting with this or other functions offered. In general, the decision on weighted or unweighted least squares can be reached on the basis of a statistical test, (see Eq. 6.2.5), or on the basis of a theoretical model.

6.2.4 Linear Least Squares Fitting in Case of Errors in Both Variables

In Fig. 6.5 three different calibration functions are given. First, the model to estimate y from (practically) error-free x values. This model is commonly used for analytical calibration in form of OLS, y^ = ax + bx x (Eq. 6.6). Another model (usually without relevance in analytical calibration because 6.2 Single Component Calibration 139 the measured values y are considered to be practically error-free) has been formulated in Eq. (6.7), x^ = ay + by y . The third model represented in Fig. 6.5, y^ = a + bx (6.39) is the so-called error in both variables model (EBV model) and has to be estimated by orthogonal least squares minimizing, that means the errors in both the dependent and the independent variable are minimized simultaneously. So, the general condition at Eq. (6.14) becomes, with Eq. (6.18): n 2 ey ! i = min (6.40) s2 + b2s2 i=1 yi xi The EBV model has to be applied if both the measured values y and the analytical value x are error-affected quantities. The calibration coefˇcients of the model at Eq. (6.39) cannot be determined directly for the general case, but only according to certain assumptions or by approximations (Danzer et al. [1995]) from which three will be given here: (1) Mandel's approximation (MA, see Mandel [1984]) % 2 2 Syy Sxx + (Sxx Syy) +4Sxy ^ bMA = (6.41) 2 Sxy

0 with Sxx, Syy, and Sxy as deˇned in Eq. (6.3 ). (2) Wald's approximation (WA, see Wald [1940]; Sharaf et al. [1986])

g n yi yj ^ i=1 j=h bWA = g n (6.42) xi xj i=1 j=h

with n being the number of calibration measurements and the index g = n=2=h 1ifn is even, and g = 1=2(n +1)=h if n is uneven. (3) Geometrical averaging (GA, see Danzer [1990]; Danzer et al. [1995]) 1 arctan bx + arctan by b^ = tan (6.43) GA 2

with bx according to Eq. (6.8) and by according to Eq. (6.10). The in- tercepts a^MA, a^WA, and a^GA are each estimated analogously to Eq. (6.9) ^ ^ ^ using bMA, bWA, and bGA instead of bx . 140 6 Calibration in Analytical Chemistry

Furthermore it should be mentioned that the ˇrst principal component p1 of a principal component analysis (PCA) gives a good approximation of the orthogonal calibration line (Danzer et al. [1995]). Principal component analysis is a useful method to extract new uncorrelated variables from a data matrix X. The mathematical fundamentals of PCA can be found in Sect. 8.3, Malinowski and Howery [1980]; Flury [1988]; Martens and Ns [1989]. The PC principle may be explained graphically as follows: In an m-dimensional data space a new coordinate system is spread by principal components in such a way that the ˇrst principle component p1 stretches in the direction of the largest variation of the data, the second one, p2,inthe direction of the largest remaining variation, orthogonal to p1, and so on. In case of calibration mean-centered data have to be used and the complete principal component solution is relevant. There exist only two principle components, p1 and p2, where the ˇrst, p1, ˇts very good the EBV calibration straight line and the second, p2, represents the orthogonal scattering of the data around the straight line. The complete principal component decomposition of the data matrix X into a score matrix P and a loading matrix P is given by X = P LT : (6.44) By means of the transposed loading matrix l l LT = 11 12 (6.45) l21 l22 the slope of the EBV (orthogonal) calibration line can be estimated according to ^ l12 bPC = (6.46) l11 ^ and a^PC again analogously to Eq. (6.9) using bPC instead of bx . Diverse procedures of EBV calibration and OLS have been compared for both simulated and real analytical data in Danzer et al. [1995]. Especially in cases when large errors in the x values (concentration) exist, EBV calibration should be used instead of OLS.

6.2.5 Statistical Tests and Validation of Calibration A calibration procedure has to be validated with regard to general and spe- ciˇc requirements under which the calibration model has been developed. For this purpose, it is important to test whether the conditions represented in Fig. 6.6 are fulˇlled. On the other hand, it is to assure by experimental studies that certain performance features (accuracy, precision, sensitivity, selectivity, speciˇcity, linearity, working range, limits of detection and of quantiˇcation, robustness, and ruggedness, see Chap. 7) fulˇl the expected requirements. 6.2 Single Component Calibration 141

Fig. 6.8. Typical plots of residual deviations: random scattering (a), systematic deviations indicating nonlinearity (b), and trumpet-like form of heteroscedasticity (c)

The underlying calibration procedure of a newly developed analytical method has to be examined by basic validation studies to determine the reliability of the method and its efˇciency in comparison with traditional methods. In order to ensure long-term stability, it is necessary to perform revalidations, which can be combined with the use of quality control charts, over meaningful time periods. Regarding calibration, it is of special importance to characterize the following features: (i) Accuracy of analytical results (ii) Precision of calibration and analytical values (iii) Calibration model (linear/nonlinear) and scope (working range, sensitivity, limits of application/detection/quantitation) Commercial software packages are usually able to represent graphically the residual errors (deviations) of a given calibration model which can be examined visually. Typical plots as shown in Fig. 6.8 may give information on the character of the residuals and therefore on the tests that have to be carried out, such as randomness, normality, linearity, homoscedasticity, etc. Accuracy. In general, the accuracy of analytical results is assured by recovery studies (Wegscheider [1996]; Danzer [1995]; Burns et al. [2002]). According to the recovery function in the general three-dimensional calibration model (see Fig. 6.3), common studies on systematic deviations (Fig. 4.3), and Eqs. (4.2) and (4.3) the following recovery formulae

xtest = ˛ + ˇxtrue (xsample = ˛ + ˇxstandard) (6.47a)

xtest(I) = ˛ + ˇxtrue(II) (6.47b) have to be tested. By means of selected standard samples (if possible, certi- ˇed reference materials) with known (\true") analyte amounts the recovery function (validation function (see Danzer [1995]; Burns et al. [2002]) has to be determined by normal linear regression4. The measures ˛ and ˇ repre- 4 It should be noted that a recovery factor obtained using a single reference material or single spike addition experiment does not indicate the absence of systematic 142 6 Calibration in Analytical Chemistry sent here the validation coefˇcients with the analytical meaning of a constant bias (intercept ˛) and a proportional bias (slope ˇ). The estimates of ˛ and ˇ can be determined analogous to Eqs. (6.8) and (6.9). By testing the null hypotheses H0: ˛ = 0 and H0: ˇ = 1 according to ˛ t^(˛) = (6.48a) s˛ ˇ t^(ˇ) = (6.48b) sˇ the absence of systematic deviations can be veriˇed. For this purpose, the estimated t^-values have to be compared with the corresponding quantiles of the t-distribution t1˛;=n2. Systematic deviations are also detected if the corresponding conˇdence intervals of the validation coefˇcients do not include 0 or 1, respectively, namely (i) An additive bias if ˛ + ˛ < 0or˛ ˛ > 0(˛>j˛j) (ii) A proportional bias if ˇ + ˇ < 1orˇ ˇ > 1(ˇ>jˇj) Another way to verify the accuracy of a method (I) under validation is the analysis of a given set of test samples with graduated analyte concentrations by both methods, (I), and an independent one (II), which is known to be accurate. The special recovery function in this case is given by Eq. (6.47b). Because both quantities, xtest(I) and xtest(II) are subject to error in this processing, EBV ˇtting according to Eqs. (6.41){(6.43) or principal component analysis (Eq. 6.46) must be applied. The test on signiˇcant deviations from ˛ = 0 and ˇ = 1 are carried out as above.

Precision. The precision of the calibration is characterized by the conˇ- ^ dence interval cnf(y i) of the estimated y values at position xi according to Eq. (6.30). In contrast, the precision of analysis is expressed by the pre- ^ ^ diction intervals prd(y i) and prd(xi), respectively, according to Eqs. (6.32) and (6.33). The precision of analytical results on the basis of experimental calibration is closely related to the adequacy of the calibration model.

Linearity. Whether the chosen linear model is adequate can be seen from the residuals ey over the x values. In Fig. 6.8a the deviations scatter randomly around the zero line indicating that the model is suitable. On the other hand, in Fig. 6.8b it can be seen that the errors show systematic deviations and even in the given case where the deviations alternate in the real way, it is indicated that the linear model is inadequate and a nonlinear model must be chosen. The hypothesis of linearity can be tested:

error or that an analytical procedure has successfully been validated (Wegscheider [1996]; Burns et al. [2002]) 6.2 Single Component Calibration 143 (a) A priori (no actual nonlinear model is considered) by comparison of the deviations of the means from the calibration line (the residual standard deviations sy:x; see Eq. (6.19)) with that of the y values from their means (sy ): p 1 2 ni(y y^i) 2 p 2 i sy:x F^ = = i=1 (6.49) 2 p ni sy 1 (y y )2 n p ij i i=1 j=1

with ni being the number of measurements in the p calibration points, see Eq. (6.13a,b). The test is carried out by comparison of the quotient (Eq. 6.49) with the corresponding quantile of the F -distribution. ^ In case that F exceeds F1˛;1=p 2;2=np a linear relationship cannot be assumed. This kind of ANOVA (Sharaf et al. [1986]) is usually implemented into software packages of regression. (b) A posteriori by comparison of the residual standard deviation of the linear model, s2 , with that of a certain nonlinear model, s2 : y:xlin y:xnon n 1 2 (yi y^i) 2 sy:x lin F^ = lin = i=1 (6.50) s2 n y:xnon 1 2 (yi y^i) non i=1

The number of degrees of freedom in case of linear models is lin = n 2orn 1, respectively, depending on whether two parameters are estimated according to Eq. (6.6) or (6.39) or only one parameter according to y^ = bx. In the nonlinear case, non results from the actual 2 model (e.g., for a quadratic equation, y^ = a + bx+ cx , non = n{ 3). A suitable test for Eq. (6.50) can also be carried out according to Mandel [1964]: s2 s2 ^ y:xlin y:xnon F = 2 (6.51) sy:xnon ^ by comparison with F1˛;1=1;2=non . In each case where F F1˛;1;2 , the linear model cannot be applied. Homoscedasticity. Unequal variances are recognizable from residual plots as in Fig. 6.8c where frequently ey is a function of x in the given trumpet-like form. In such a case, the test of homoscedasticity can be carried out in a Hartley ^ 2 2 simple way by means of the test (Fmax test), Fmax = smax=smin, see Sect. 4.3.4 (1). 144 6 Calibration in Analytical Chemistry In cases in which the situation is more obscure as represented in Fig. 6.8c, the Bartlett test of homogeneity of variances (see Sect. 4.3.4 (3)) has to be applied. The test statistic ^2 has to be compared with the critical value and 2 2 2 2 2 the null hypothesis H0: 1 = 2 = = p must be rejected if ^ 1˛; . In this case, instead of normal OLS calibration, the weighted calibration has to be chosen. For WLS the dependence of the variance of the measure- 2 ment values, sy , on the analytical values x(and therefore on the measured values y , too) has to be determined in preliminary studies. Then the result- 2 ing variance function, sy = f (y ), can be entered for weighting directly in 2 corresponding calibration programs, e.g. in the form w =1=sy =1=1=y ; see Sect. 6.2.3. Sometimes, as an alternative to the weighted calibration, it is recommended to restrict the working range to those partial ranges in which homoscedasticity can be supposed. Considering the relatively frequent occurrence of heteroscedastic calibration data, however, in analytical practice, the possibility of applying WLS should, in general, be taken into account to a stronger extent. In this way, more reliable analytical results can be obtained with a higher precision which is constant to a large extent and covers the whole working range.

6.2.6 Alternative Calibration Procedures

If the basic conditions for the use of least squares ˇtting are not fulˇlled (Fig. 6.6), especially if strongly deviating calibration points appear (\outliers" or, more exactly, leverage points), the OLS method fails, i.e., the estimated calibration are biased and, therefore, are not representative for the relation between x and y . Whereas normality of the measured values can be frequently obtained by a suitable transformation, especially in the case of outlying calibration points, robust calibration has to be applied (Rousseeuw and Leroy [1987]; Danzer [1989] Danzer and Currie [1998]).

Robust calibration. The Gaussian OLS criterion according to Eq. (6.16) is strongly sensitive against outliers. Therefore, robust methods of ˇtting have been developed following two strategies (Rousseeuw and Leroy [1987]): (i) Recognition and selection of outliers and ˇtting of the remaining data by OLS (outlier diagnostics) (ii) Fitting of only the representative data by means of robust techniques (robust regression in the stricter sense) It has been known for a long time (Edgeworth [1887]) that minimizing of the linear sums of deviations n ! eyi = min (6.52) i=1 6.2 Single Component Calibration 145

Fig. 6.9. Schematic representation of some of the slopes bij between eight calibration points5 including an outlier (black marked) is more insensitive against outliers than that of squared sums according to Eq. (6.16). Robust regression techniques are also based on linear deviation criteria and use special minimization functions (Huber [1981]) and inuence functions (Hampel [1980]; Hampel et al. [1986]): n ! q eyi = min (6.53) i=1 On the other hand, least trimmed squares (LTS) can be used to estimate the calibration parameters: that points which produce the (n h) largest deviation squares are rejected and the remaining h points are used for the LTS ˇtting: h ! e2 = min (6.54) yi i=1 Another way is the robust parameter estimation on the basis of median statistics (see Sect. 4.1.2; Danzer [1989]; Danzer and Currie [1998]). For this, all possible slopes between all the calibration points bij =(yj yi)=(xj xi) for j>iare calculated. After arranging the bij according to increasing values, the average slope can be estimated as the median by ~ b = medfbijg (6.55) where medfbijg is determined according to Eq. (4.22). The principle of the procedure is illustrated in Fig. 6.9.

5 The total number nb of all the single slopes bijis given by the combinations of n n calibration points to the 2nd class: n = C 2 = = 1 n(n 1), i.e., in this case b n 2 2 nb =28. 146 6 Calibration in Analytical Chemistry The intercept a~ will then be obtained by ~ a~ = medfyi bxig (6.56) Estimates of the variance and uncertainty intervals in robust calibration can be taken from the literature (Huber [1981]; Rousseeuw and Leroy [1987]). Robust calibration corresponds in most cases to the problem of outlying calibration points (leverage points). In consideration of that, attention must be directed to the linearity of the relationship in general and the randomness of the residuals.

Calibration by Standard Addition Method (SAM). When matrix effects appear or are to be expected and matrix-matched calibration samples are not available, the standard addition method (SAM) can be the calibration method of choice. Especially in the case of biochemical and environmental systems and generally in ultra trace analysis, SAM is frequently applied. By addition of standard solutions to the sample a similar behaviour of the calibration set and the sample is created provided that the analyte is added in form of same species. The model of standard addition is based on the prerequisite that blanks do not appear, y = bx, or can be eliminated, y a = ynet = bx:

y0 = bx0 (6.57) where y0 is the measured value of the unspiked test sample and x0 the initial content of the analyte. Known amounts xi of the analyte are added to the sample; in doing so it is recommendable to use equimolar amounts of xi in the range of

x1 x0;x2 =2x1;:::;xp = px1 where frequently p = 3 or 4 is used. Therefore, some ideas about the initial concentration x0 should exist. The principle of SAM is illustrated in Fig. 6.10. The standard addition calibration function is estimated by least squares ˇtting. The slope b (sensitivity of the SAM) will be obtained according to

y^p y^0 b = (6.58) xp This procedure is justiˇed if the sensitivity of the determination of the species in the sample is the same as of the species added:

y y0 ! y^p y^0 0 b(AD) = = = b(SA) = (6.58 ) x x0 xp The proceeding is simpliˇed for such cases in which the volume of the added analyte amounts can be neglected in comparison with that of the initial sample (solution). Otherwise, a procedure using volume-corrected measurement values has to be applied (see Sharaf et al. [1986]). 6.2 Single Component Calibration 147

Fig. 6.10. Calibration by the standard addition method (SAM)

The estimation of the analyte content x is given by the SA analytical function (the inverse function of Eq. (6.57)) whose extrapolation for y =0 yields y^0 xp y^0 x0 = = (6.59) b y^p y^0 with the conˇdence interval " ! xp 2 sy:x t1˛; 1 x0 2 cnf (x0)=x0 ˙ + (6.60) b n Sxx This conˇdence interval is wider as compared to normal calibration (see Eq. 6.33) because of the extrapolation to xi = x0. The number of calibra- p tion measurements n results from n0 + p ni or n0 + ni, respectively, where n0 is the number of measurements of y0. When a blank appears, it has to be estimated from a sufˇciently large number of blank measurements and the measured values must be corrected in this respect. To ensure the adequateness of the SA calibration model, p 2 additions should be carried out. Only in the case when it is deˇnitely known that the linear model holds true, then one single addition (ni times repeated) may be carried out. In general, linearity can be tested according to Eqs. (6.49){(6.51). Although standard addition calibration is an unreliable method if linearity in the range x

Fig. 6.11. Calibration lines by using several CRM samples (S: soil BHB-1, T: Tibet soil GBW 8302, R: river sediment CRM 320, Ga: Gabbro MRG-1, A: Andesite AGV-1, Gr: Granodiorite GSP-1) and varying the sample weight (Zn 307.6 nm)

(sample solution, sample vapour, plasma etc), as it is expressed, e.g., by the general law of absorption, y = y0 exp(kN). In most cases, conditions of the analytical procedures are ˇxed which provide the direct connection between the signal intensity and the analyte content (e.g. constant measuring volumina or sample weights). Not in all cases can such constant conditions be realized. In Solid Sam- pling AAS (SS-AAS), calibration is commonly carried out by using only one suitable calibration sample (frequently a certiˇed reference material, CRM) and varying its weight. The absolute analyte amount of the analyte is then used as the independent quantity and calibrated vs the dependent signal intensity (peak height or area, respectively). The second variant, the use of several CRMs with varying analyte content, can only rarely applied because of high costs of CRMs and matrix inuences that will be distinct under the given conditions. By combination of both calibration variants it can be shown (Danzer et al. [1998]) that strong matrix effects appear which are represented by the different sensitivities in Fig. 6.11. To overcome these matrix inuences, other calibration strategies can be used (Danzer et al. [1998]). According to one of them, a specialized three- dimensional calibration model, the relation between signal intensity and both sample weight and content is evaluated. The relationship between the three quantities, y = f (x;w) is demonstrated in Fig. 6.12 by the example of Zn determination represented above. Both planar and curved surfaces can be ˇtted by statistical software packages like Statistica [1993]. In the general case, the 3-D calibration surface is given by 6.2 Single Component Calibration 149

Fig. 6.12. Three-dimensional calibration plot for the zinc determination using a plain model for the relationship peak height vs zinc content and sample weight

2 2 y = a + b1x + b2w + b3x + b4xw + b5w (6.61) where y is the signal intensity (peak height or area, respectively), x the analyte content, w the sample weight, and a as well as the bi are the 3D calibration coefˇcients. Upper and lower conˇdence planes can be estimated by bootstrapping (Efron [1982]; Efron and Tibshirani [1993]) as described in detail by Nimmerfall and Schron [2001].

Calibration by means of results of another method. In life science and other ˇelds there can exist cases in which no real calibration samples are available. That may concern substances of metabolism in living organisms like cholesterol and glucose in blood. Intensiˇed metabolic control, especially in case of diabetes, demands minimal-invasive or non-invasive methods of analytical measurement. For this goal, a method has been developed to measure the blood glucose content in vivo, in direct contact with the skin, by means of diffuse reection near infrared (NIR) spectroscopy on the basis of multivariate calibration and neural networks (Muller et al. [1997]; Fischbacher et al. [1997]; Danzer et al. [1998]). Because no patients with any \standard blood glucose value" are available in principle, a method of indirect calibration has 150 6 Calibration in Analytical Chemistry

Fig. 6.13. Proˇle of blood glucose concentration for a single test person; # taking of carbohy- drates/g, " insulin injection/IU

Fig. 6.14. Indirect calibration of NIR measurements vs BGA measurements for several test persons to be applied. For this aim, type-1 diabetes patients (involved into an inten- siˇed conventional insulin therapy) have been selected in which variations of blood glucose values were provoked by subcutaneous or intramuscular injections of 2{5 units of regular insulin and by carbohydrate supply. While recording the NIR spectra at the right hand ˇnger, blood samples were taken from a cannula placed in the dorsum of the left hand. The reference concentrations were measured by a Beckman Glucose Analyzer (BGA). A typical blood glucose proˇle obtained in this way is represented in Fig. 6.13 (Muller et al. [1997]; Fischbacher et al. [1997]). On the other hand, Fig. 6.14 shows the calibration of the NIR measurements vs the BGA reference measurements. In contrast to the common calibration procedure measuring y = f (xstandard) as shown in the front plane of Fig. (6.3a,b), the glucose reference calibration takes place in the base area of the three-dimensional model, xsample = f (xstandard); see Sect. 6.1.2. Calibration with imprecise signals and analytical values (concentrations) based on fuzzy theory has been dealt with by Otto and Bandemer [1986]. 6.2 Single Component Calibration 151 6.2.7 Nonlinear Calibration

Nonlinear relationships between analytical quantities and measured values results from different reasons. In most cases, the relating natural laws can determine the nonlinearity. On the other hand, factors and conditions can inuence an analytical system in such a way that relationships which are originally linear can become curved. Absorption of electro-magnetic radiation is an example that can illustrate both phenomena. The Lambert-Beer law of absorption describes a nonlinear dependence of the radiation intensity after the absorption step, I, on the number of absorbing species N , namely I = I0 exp(kN). I0 is the incident intensity and k an absorption coefˇcient. It is well-known that this relationship mostly is used in its linearized form, the so-called Beer law, A = log (I0=I)) = k N = " l c with A absorbance, k absorption coefˇcient, " molar absorptivity, l length of the absorbing cell, and c concentration of the absorbing species. Beer's law usually is only valid for dilute solutions (c 0:1 mol/L). At higher concentrations deviations from the linear calibration curve may occur. Because nonlinear calibration needs higher expense both in experimental and computational respects, linear models are mostly preferred.

Nonlinear calibration is carried out by nonlinear regression where two types have to be distinguished: (1) real (intrinsic) nonlinear regression and (2) quasilinear (intrinsic linear) regression. The latter is characterized by the fact that only the data but not the regression parameters are nonlinear. Typical examples are polynomials and trigonometric functions. When nonlinearity is detected, either by visual inspection, test on linearity, or logical reasons, the following possibilities can be used: (i) Transformation of the analytical values and/or measured values

xtrans = f (x);ytrans = f (y ) (6.62)

and modelling of the linear function

ytrans = a + b xtrans (6.63)

(ii) Modelling of a nonlinear relationship between y and x by means of a suitable function, some of them are given in Table 6.2. (iii) Multi-range calibration corresponding to a multi-range regression; see, e.g., Sachs [1992]. Several working ranges are ˇtted by diverse calibration straight lines where in analytical practice the use of only two ranges is customary. (iv) Spline functions ˇt small intervals of the calibration function by polynomials of low (2nd or 3rd) order under the condition that the resulting overall curve represents a continuous function (Wold [1974]). 152 6 Calibration in Analytical Chemistry

Table 6.2. Suitable nonlinear calibration functions, their normal equations and sensitivity functions

Calibration function Normal equations Sensitivity function S(x) y = a + a x + a x2 a n + a x + a x2 = y 0 1 2 0 1 2 dy a x + a x2 + a x3 = xy = a1 +2a2x 0 1 2 dx a x2 + a x3 + a x4 = x2y 0 1 2 y = a + a x + a x2 + a x3 a n + a x + a x2 + a x3 = y 0 1 2 3 0 1 2 3 2 3 4 a0 x + a1 x + a2 x + a3 x = xy dy 2 = a1 +2a2x +3a3x a x2 + a x3 + a x4 + a x5 = x2y dx 0 1 2 3 a x3 + a x4 + a x5 + a x6 = x3y 0 1 2 3 y = a0 + a1 lg x a0n + a1 lg x = y dy 1 1 = lg e = a lg x + a (lg x)2 = (y lg x) dx x x ln 10 0 1 lg y = a0 + a1 lg x a0n + a1 lg x = lg y d lg y = a a lg x + a (lg x)2 = (lg x lg y ) d lg x 1 0 1 y = a bx corresponding to n lg a +lgb x = lg y dy = bx ln b lg y =lga + x lg b lg a x +lgb x2 = (x lg y ) dx

In general, from nonlinear calibrations result variable sensitivities expressed by sensitivity functions S(x): dy df (x) S(x)=y 0 = f 0(x)= = (6.64) dx dx For the purpose of comparisons, the sensitivity of the centre of the calibration curve S(x) having the character of an average sensitivity can be used. Sensitivity functions corresponding to diverse calibration functions are given in Table 6.2.

6.3 Multisignal Calibration

By diverse spectroscopic techniques not only a single signal is generated by given species but several their number can be lower (XRF, MS) or higher (OES). Whereas especially in OES several spectral lines are used for identiˇ- cation and qualitative analysis, quantitative analysis of a given analyte is carried out by means of only one single line. In doing so, mostly the most sensitive, undisturbed line is selected whereas all the other signals are ignored. It has been shown (Danzer and Wagner [1993]; Danzer and Venth [1994]; Venth et al. [1996]) that the reliability of quantitative analyses can be increased when several signals are used for calibration and evaluation. From theoretical considerations it is expected that a multivariate evaluation increases both the sensitivity and the precision and, therefore, the detection power, too. Multisignal evaluation is carried out by means of Principal Component Analysis (PCA) or (Partial Least Squares (PLS) regression. The fundamentals 6.3 Multisignal Calibration 153 A B 2.0 Mnpc y 1.6 A2 y Apc 1.2 Mn1 Mn2,Mn3 0.8 Mn4,Mn5 0.4 y

0 y 0 0.1 0.2 0.3 0.4 A1 x relMn C y Si1 Fig. 6.15. A Principle of the derivation of a y Si pc latent intensity yApc from original intensities B yA1 and yA2 . Sensitivity of the pc intensity in comparison to the ˇve original intensities of the lines Mn 279.8 nm, Mn 280.1 nm, Mn 293.3 nm, Mn 403.3 nm, and Mn 403.4 nm (Danzer and Venth [1994]). C pc intensity ySi y 3 Si 2 of Si resulting from Si 251.5 nm, Si 252.9 nm and Si 288.2 nm (Danzer et al. [1991]) of these chemometric methods can be found in Chapter 8: Dillon and Goldstein [1984] Martens and Ns [1989]; Danzer and Venth [1994]; Venth et al. [1996]. The principle of multisignal calibration will be represented here in a more illustrative way. Starting with two signal intensities of one and the same analyte A, namely yA1 and yA2 . Plotting yA1 vs yA2 as done in Fig. 6.15A, it can be seen that the resulting (latent) intensity yApc is larger then the both original values of the same species. Figure (6.15B,C) shows the results of experimental studies that conˇrm the theoretical expectations. Because the ˇrst principal component is always situated in direction of the largest variation of the data, the intensity of ypc should be increased compared with the original intensities from which it was calculated, namely theoretically according to the n-dimensional Pythagoras' principle n 2 ypcmax = In (6.65) i=1 According to the fundamentals of PCA, simultaneously the precision should be improved because the noise is preferably to ˇnd in the higher pcsas shown by simulations and experimentally in Danzer and Wagner [1993], OES determination of Mn, Ni, and Cr in steel (Danzer and Venth [1994]) 154 6 Calibration in Analytical Chemistry

Table 6.3. Spectral interferences in ICP-MS of the ˇve most abundant cadmium isotopes relevant for the cadmium determination in Mo-Zr alloys; additional Cd isotopes: 106Cd 1.25%, 108Cd 0.89%, 116Cd 7.49%

Mass Abundance Molecule ion Abundance Isobaric Abundance (%) interferences (%) interferences (%) ZrO 17.49 110 12.49 ZrF 11.20 Pd 11.72 MoO 9.31 ZrOH 17.31 ZrF 17.10 111 12.80 MoO 15.86 MoF 14.80 MoOH 9.10 MoO 16.68 112 24.13 MoOH 15.66 Sn 0.97 ZrO 2.83 ZrF 17.50 MoOH 16.46 In 4.30 113 12.22 MoO 9.58 MoF 9.3 ZrOH 2,79 MoO 24.07 114 28.73 MoF 15.90 Sn 0.65 MoOH 9.43 and ICP-MS determination of Sb, Cd, In, Pd, Ag, and Sn in Mo-Zr alloys (Venth et al. [1996]). The advantage of multisignal calibration was shown impressively in case of cadmium. Its determination on the basis of single signal evaluation is practically impossible because each signal of the Cd isotopes is disturbed as shown in Table 6.3. By means of the most sensitive signal Cd-114 only a relative uncertainty of 22% could be observed. In contrast, PCA calibration including the ˇve isotopes given in Table 6.3 yields an uncertainty of 0.6% with nearly 100% recovery. This example demonstrates that not only undisturbed signals can be included in the calibration but also disturbed ones. 6.4 Multicomponent Calibration 155 6.4 Multicomponent Calibration

The most important techniques in multicomponent analysis are spectroscopy (of radiation of various wavelengths as well as of particles) and chromatography. Spectroscopic and chromatographic methods are able to analyze diverse species in a more or less selective way. For the determination of n species Qi (i =1; 2;:::;n), see 2.3, at least n signals must be measured, which should be well-separated in the ideal case. In analytical practice, the situation can be different as shown in Fig. 6.16; see Eckschlager and Danzer [1994]; Danzer et al. [2004]. The given detail of a spectrum may show either well-separated signals as represented in (a) or signals that are overlapped to different degree; see (b) and (c).

⎛ ⎞ a11 00 ⎝ ⎠ A = 0 a22 0 (6.66) 00a33

⎛ ⎞ a11 a12 0 ⎝ ⎠ A = a21 a22 a23 (6.67) 0 a32 a33

⎛ ⎞ a11 a12 a13 ⎝ ⎠ A = a21 a22 a23 (6.68) a31 a32 a33

Fig. 6.16. Evaluation of multicomponent analysis in cases of various signal relations: well- separated (a), moderately overlapped (b), and strongly overlapped (c) in form of spectra (left) and relevant matrices { Eqs. (6.66) to (6.68) (right); Q1, Q2;Q3 are different species (analytes), z1, z2, z3 wavelengths at which the intensities y1, y2, y3 are measured, A are the matrixes of absorbances and aij absorption coefˇcient of species i at wavelength j 156 6 Calibration in Analytical Chemistry In case (a) each species can be calibrated and evaluated independently from the other. In this fully selective case, the following equation system corresponds to the matrix A in Eq. (6.66):

y1 = a10 + a11x1 + e1 y2 = a20 + a22x2 + e2 . (6.69) . ym = am0 + amnxn + em where the number of measured signal absorbances, m, usually is equal to the number of species, n. On the other hand, case (b), in which neighbouring signals overlap to a certain degree, can be handled by means of multiple linear calibration provided that the absorbances yi are additive and related signal maxima can be measured for each species:

y1 = a10 + a11x1 + a12x2 + + a1nxn + e1 y2 = a20 + a21x1 + a22x2 + + a2nxn + e2 . (6.70a) . ym = am0 + am1x1 + am2x2 + + amnxn + em or, in matrix form,

y = Ax + ey (6.70b) Multicomponent systems of the kind shown in Fig. 6.16b can be calibrated with a high degree of reliability when the preconditions mentioned above are valid. The uncertainties ei contain both deviations from the model and random errors. In the case of strongly overlapped signals, see Fig. 6.16c, multiple linear calibration cannot be used for the following reasons: (i) In real analytical systems, not all the sample constituents are known. In such cases, an alternative way may be the application of the inverse calibration model of Eq. (6.70a):

x = Ya+ ex (6.71)

where Y is a matrix of spectra with m given wavelengths for n component mixtures in which other variations like baseline effects must be included. The vector of the sensitivity coefˇcients a can be estimated by a^ =(YT Y)1 YT x (6.72)6

(ii) In principle, for spectra like that in Fig. 6.16c, multicollinearities have to be expected. That means that overlapping signal functions and consequently the resulting sum curve are correlated and the measured absorbance values at the respective wavelengths are not independent

6 YT is the transposed of the matrix Y. 6.4 Multicomponent Calibration 157

Fig. 6.17. Multivariate calibration methods (according to Danzer et al. [2001, 2004])

from each other. Therefore, Eq. (6.72) becomes unstable and other methods of estimation of the a values must be used. These methods use overdetermined equation systems as their basis, such as a11 a12 a13 a1m A = a21 a22 a23 a2m (6.73) a31 a32 a33 anm

instead of Eq. (6.68). The number of wavelengths, m (sensors, detecting channels, respectively), is usually much higher than the number of species n (here n = 3). The estimation of the calibration coefˇcients is then carried out by multivariate calibration. Depending on whether the spectra Y are calibrated as dependent on analyte amounts X or conversely, different methods of multicomponent calibration, as represented in Fig. 6.17, can be applied.

6.4.1 Classical Multivariate Calibration

The classical multivariate calibration represents the transition of common single component analysis from one dependent variable y (measured value) to m dependent variables (e.g., wavelengths or sensors) which can be simultaneously included in the calibration model. The classical linear calibration (Danzer and Currie [1998]; Danzer et al. [2004]) is therefore represented by the generalized matrix relation Y = XA (6.74) where Y is the (p m) matrix of dependent variables (e.g., absorbances at m wavelengths or responses at m sensors), p is the number of calibration 158 6 Calibration in Analytical Chemistry standards (calibration mixtures) that is identical with the number of spectra or similar measured records, X is the (p n) matrix of independent variables (e.g., the concentration of the n components), and A is the (n m) matrix of the calibration coefˇcients, often called the sensitivity matrix (Kaiser [1972]; Martens and Ns [1989]; Danzer [2001]). The rows of the matrix A correspond to the spectra of the pure components, which can be directly measured or indirectly estimated. Direct calibration can be applied when the calibration coefˇcients are known, otherwise { in case of indirect calibration { the calibration co- efˇcients are computed by means of experimentally estimated spectra- concentrations relations. Classical calibration procedure can only be applied when all the species that contribute to the form of the spectra are known and can be included into the calibration. Additionally, there is the constraint that no interactions between the analytes and other species (e.g. solvent) or effects (e.g. of temperature) should occur. The analytical values (concentrations) are estimated by x^ = YA+ (6.75) where A+ is the so-called Moore-Penrose generalized pseudo inverse (Moore [1920]; Penrose [1955]) A+=(ATA)1AT (6.76) with the same dimensions (m p ) as the transposed matrix. In case of baseline shift, the sensitivity matrix in Eqs. (6.74) and (6.75) must be complemented by a vector 1: ⎛ ⎞ 1 a11 a1m 1 ⎝ . . . ⎠ A =( A)= . . . (6.77) 1 ap1 apm Instead of the addition of the 1-vector the calibration data may be centered (yi y and xi x, respectively). Even if the spectra of the pure species cannot be measured directly then the A-matrix can be estimated indirectly from the spectra provided that all components of the analytical system are known: A^ = (XTX)1XTY (6.78)7 For inversion of the matrix XT X it is necessary that a sufˇcient number of spectra for different concentration steps have been measured. The concentration vectors must vary independently from each other. For this reason, experimental design (see, e.g., Deming and Morgan [1993]) can be

7 Instead of the symbol A and the term sensitivity matrix also the symbol K (matrix of calibration coefˇcients, matrix of linear response constants etc) is used. Because of the direct metrological and analytical meaning of the sensitivities aij in the A-matrix the term sensitivity matrix is preferred. 6.4 Multicomponent Calibration 159 used. In the case that the preparation of samples of deˇned composition is impossible, then the samples should be selected as representative and as uncorrelated as possible (natural design, see Martens and Ns [1989]). The prediction of analytical values X according to the classical indirect calibration model follows Eq. (6.75): + X^ = Y A^ (6.79) The desired independence between the variables of the different analytical signals corresponds directly with the selectivity of the analytical system (Kaiser [1972]; Danzer [2001], and Sect. 7.3). In case of multivariate calibration, the selectivity is characterized by means of the condition number cond(A)=kAkkA1k (6.80) 1 where kAk is the matrix norm of A and kA kpthe norm of the inverse matrix. The matrix norm of A is calculated from max , the square root of 1 the largest eigenvalue max, and the norm of A from the reciprocal square root of the lowest eigenvalue min: p cond(A)= p max (6.81) min Equation (6.79) is valid for exactly determined systems (m = n). In case of overdetermined systems, m>n, the condition number is given by % cond(A)= cond(ATA) (6.82) If systems are well-conditioned the selectivity is expressed by condition numbers close to 1. The uncertainty in multivariate calibration is characterized with respect to the evaluation functions at Eqs. (6.75) and (6.79). The prediction of a row vector x of dimension n from a row vector y of dimension m results from x = yAT(AAT)1 (6.83) The relative uncertainty for the prediction of the x-values can be estimated by kıxk kıyk kıAk = cond A + (6.84) kxk kyk kAk where kıyk=kyk is the relative uncertainty of the y -values (error of measurement) and kıAk=kAk the relative uncertainty of the estimation of A (modeling error). The condition number is calculated from Eqs. (6.80){(6.82).

6.4.2 Inverse Calibration The classical direct or indirect calibration is carried out by OLS minimization according to Gauss. Error-free analytical values x are assumed or at 160 6 Calibration in Analytical Chemistry least that the errors in x are very small compared with those of the y -values (Danzer and Currie [1998]). Additionally all the components in the analytical system must be known and included in the calibration. If these preconditions are not fulˇlled the inverse calibration must be applied. The inverse calibration regresses the analytical values (concentrations), x, on the measured values, y . Although with it a prerequisite of the Gaussian least squares minimization is violated because the y -values are not error- free, it has been proved that predictions with inverse calibration are more precise than those with the classical calibration (Centner et al. [1998]). This holds true particularly for multivariate inverse calibration. In chemometrics, the inverse calibration model is also denoted as the P-matrix model (the dimension of P is m n): X = YP (6.85) The calibration coefˇcients are elements of the matrix P which can be estimated by P^ = Y+ X =(YTY)1YTX (6.86) The analysis of an unknown sample is carried out by multiplication of the measured spectrum y by the P-matrix x^ = y P^ (6.87) In the case that the original variables, the measured values y, are used for inverse calibration, there are no signiˇcant advantages of the procedure apart from the fact that no second matrix inversion has to be carried out in the analysis step; see Eq. (6.87). On the contrary, it is disadvantageous that the calibration coefˇcients (elements of the P-matrix) do not have any physical meaning because they do not reect the spectra of the single species. In addition, multicollinearities may appear which can make inversion of the Y-matrix difˇcult; see Eq. (6.86). On the other hand, when latent variables instead of the original variables are used in inverse calibration then powerful methods of multivariate calibration arise which are frequently used in multispecies analysis and single species analysis in multispecies systems. These so-called \soft modeling methods" are based, like the P-matrix, on the inverse calibration model by which the analytical values are regressed on the spectral data: X = YB (6.88) where B is the (m n)-matrix of calibration coefˇcients, in concrete terms the matrix of B-coefˇcients. In contrast to the P-matrix, not all the dimensions of the spectra (the Y-matrix) are used but only those that are signiˇcant are realized by certain principal components. Therefore, the estimation of the matrix of B-coefˇcients can be carried out by PCR (Principal Component Regression) or PLS (Partial Least Squares) Regression. Both PCR and PLS form latent variables T (principal components, factors) from the original variables, viz., from the matrix of measured values according to 6.4 Multicomponent Calibration 161 T Y = TL + EY (6.89) where T is the factor (score) matrix and L the loading matrix with the dimension m n; EY is the matrix of non-signiˇcant factors which is regarded as an error matrix. Additionally, in PLS the matrix of analytical values (e.g. concentrations) is decomposed in the same way: T X = TQ + EX (6.90) PCR and PLS have in common the following steps: (1) Estimation of a weight matrix (eigenvalues) V (from Y in PCR and from Y and X in PLS) (2) Calculation of the factor matrix T = ZV by means of the standardized variables Z (3) Calculation of the matrices P and Q according to PT = T+Y (6.91) QT = T+X (6.92) In PCR the calibration coefˇcients (B-matrix) are estimated column by column according to b^ = VQT (6.93) and ^ b0 = x Y b^ (6.94) The prediction then is carried out by ^ x^ = Y b^ + b0 (6.95) The signiˇcance and non-signiˇcance of principal components are decided on the basis of the variance that is explained by each them. Normally, in analytical methods the main variance is caused by the analyte concentration. But sometimes properties of the sample, such as moisture or surface roughness, or effects of the measuring procedure such as spectral baselines or scattered light, can exceed the effect of analyte concentration. Therefore, additional tests should be made as to what degree the principal components postulated to be non-signiˇcant by the software are correlated with the analytical values. Principal components which are highly correlated with the variable of interest (e.g., concentration) should be included in the calibration procedure notwithstanding their share in the variance. In PLS both the matrices of measured values Y and analytical values X T are decomposed according to Eqs. (6.89) and (6.90): Y = TP + EY and T X = TQ + EX and thus relations between spectra and concentrations are considered from the outset. The B-matrix of calibration coefˇcients is estimated by B^ = V(PT V)1QT (6.96) 162 6 Calibration in Analytical Chemistry Because the Y-matrix and X-matrix are interdependently decomposed the B-matrix ˇts better and more robust than in PCR the calibration. The evaluation is carried out by Eq. (6.88) according to X^ = YB^. The application of PLS to only one y -variable is denoted as PLS 1. When several y -variables are considered in the form of a matrix the procedure is denoted PLS 2 (Manne [1987]; Hskuldsson [1988]: Martens and Ns [1989]: Faber and Kowalski [1997a, b]).

6.4.3 Validation of Multivariate Calibration

The reliability of multispecies analysis has to be validated according to the usual criteria: selectivity, accuracy (trueness) and precision, conˇdence and prediction intervals and, calculated from these, multivariate critical values and limits of detection. In multivariate calibration collinearities of variables caused by correlated concentrations in calibration samples should be avoided. Therefore, the composition of the calibration mixtures should not be varied randomly but by principles of experimental design (Deming and Morgan [1993]; Morgan [1991]).

Selectivity. In general, selectivity of analytical multicomponent systems can be expressed qualitatively (Vessman et al. [2001]) and estimated quantitatively according to a statement of Kaiser [1972] and advanced models (Danzer [2001]). In multivariate calibration, selectivity is mostly quantiˇed by the condition number; see Eqs. (6.80){(6.82). Unfortunately, the condition number does not consider the concentrations of the species and gives therefore only an aid to orientation of maximum expectable analytical errors. Inclusion of the concentrations of calibration standards into selectivity models makes it possible to derive multivariate limits of detection.

Precision. The uncertainty of calibration and prediction of unknown concentrations are expressed by the standard error of calibration (SEC), deˇned as n 2 y y (cs) icalc itrue s^ = i=1 (6.97) cal n and the standard error of prediction (SEP), deˇned as n 2 y y (ts) icalc itrue s^ = i=1 (6.98) pred n where y (cs) are the true values of the calibration samples (cs, e.g. standards), itrue y (ts) the true values of test samples (ts) with which the prediction power itrue 6.4 Multicomponent Calibration 163

independently is estimated, and yicalc are the respective y -values calculated by the model. Another measure for the precision of multivariate calibration is the so- called PRESS-value (predictive residual sum of squares, see Frank and Todeschini [1994]), deˇned as n n 2 2 2 sres = ei = (yicalc yitrue ) (6.99) i=1 i=1 It can be calculated as usual for SEP, see Eq. (6.98) by use of test samples. It is also possible to estimate the PRESS-value on the basis of standard samples only applying cross validation by means of the so-called hat matrix H (Faber and Kowalski [1997a, b]; Frank and Todeschini [1994]): H = X(XT X)1XT (6.100) The n n hat matrix transforms the vector of the measured y -values to the vector of the estimated-values. An element hij of the hat matrix is calculated by T T 1 hij = xi (X X) xj (6.101) From the elements of the hat matrix some important relations can be derived, e.g. the rank of the X-matrix from the sum of the signiˇcant diagonal elements of the hat matrix n rank(X)= hii (6.102) i=1 (the rank of the hat matrix is equal to its trace) and the residuals e^ = y y^ = y X(XT X)1XT y =[I X(XT X)1XT ] y =[I H] y (6.103) The residuals can be calculated from a given set of calibration samples in a different way. Cross validation is an important procedure to estimate a realistic prediction error like PRESS. The data for k samples are removed from the data matrix and then predicted by the model. The residual errors of prediction of cross-validation in this case are given by

e^(k) e(cv) = (6.104) 1 hkk The PRESS value of cross-validation is given by the sum of all the k variations 2 2 s(cv)res = e^(cv) (6.105)

Prediction limits for the estimation of an unknown concentration xi can be calculated. The calculation depends on the speciˇc multivariate calibration model

xi;pred = xî ˙ x^ = xî ˙ sx t1˛; (6.106) 164 6 Calibration in Analytical Chemistry where xi;pred is the mean of the predicted unknown concentration and sx is the standard deviation of prediction estimated from n 2 (yi yî) s2 = i=1 (6.107) x m p with t1˛; being the Student-t statistic for degrees of freedom at the 2 1 ˛ conˇdence level. The variance sx depends on the number of sensors or wavelengths, m, the number of species, n, the number of parameters p and a factor which takes the form =(BT B)1 (6.108) in the case of classical multivariate calibration. For inverse calibration, T T 1 =1+y0 (Y Y) y0 (6.109) and for cross-validation when the leverage values are applied in calibration,

=1+hkk (6.110)

Trueness. Absence of systematic errors can be tested traditionally by means of recovery functions; see Sect. 6.1.2, Fig. 6.3C; Burns et al. [2002]. For this reason the concentration estimated by the model, x^, is compared with the true concentration value, xtrue, by a regression model

x^ = ˛ + ˇxtrue (6.111) where the xtrue can be the known values of an independent set of test samples or reference values estimated on the same samples by means of an independent method which yields true values as is well known. The regression coefˇcients have to be ˛ = 0 and ˇ = 1 where values outside of the conˇdence interval ˙˛ indicate additive (constant) systematic errors and values exceeding the conˇdence interval 1˙ˇ upwards or downwards show proportional systematic errors. By means of recovery studies both accuracy can be tested and precision can be estimated.

Multivariate limit of detection. Starting from a model like Eq. (6.74) in which the background vector y0 is included: 8 y = AX + y0 (6.112) and its solution for the concentration vector + x = A (y y0) (6.113) Bauer et al. [1991a] derived the following propagation of uncertainty for x + + + + dx = A (dY dY0 )X x +dXX x + A (dy dy0 ) (6.114) 8 To avoid confusion, A is used here as symbol for the sensitivity matrix; Bauer et al. [1991a, b] use S for this purpose and B for the background vector (here y0). 6.5 Calibration by Artiˇcial Neural Networks 165 where the d-vectors and -matrixes are errors of the respective quantities. From Eq. (6.114) the variances s2 of the contents x are obtained: xk k P N K s2 = (A+X+x)2(s2 + s2 ) xk Y Y0 p =1 n=1 k=1 (6.115) N K P + (X+x)2s2 + (A+)2(s2 + s2 ) : X y y0 n k p

By means of these variances the limits of detection xkLD of the k analytes under investigation can be estimated in analogy to the univariate case (see

Sect. 7.5), namely on the basis of the critical values ynetk;c of the net signals their vector is denoted yk;c in the last term of Eq. (6.116b):

xkLD = u˛ sxk (6.116a) P N K N K P (A+X+x)2(s2 + s2 )+ (X+x)2s2 + (A+)2(s2 + s2 ) xkLD = u˛ Y Y0 X yk;c y0 (6.116b) p =1 n=1 k=1 n k p A detailed derivation can be found in Bauer et al. [1991b]. The limit of detection according to Eq. (6.116a) corresponds to Kaiser's so-called 3 criterion; see Sect. 7.5., Lorber and Kowalski [1988] as well as Faber and Kowalski [1997b] take into account errors of the ˇrst and second kind. The multivariate detection limits are estimated then in analogy to the univariate limits being twice the 3-limit (with u˛ = uˇ); see Sect. 7.5 and Ehrlich and Danzer [2006]).

6.5 Calibration by Artiˇcial Neural Networks

Neural networks are systems of information processing that consists of a large number of units which transmit information to another by means of directed connections activating other units. Artiˇcial neural networks (ANN) are simpliˇed and idealised replicas of biological neural networks. An ANN can be considered as a parallel machine that uses a large number of processor units, the neurons, with a high degree of connectivity (Zupan and Gasteiger [1991]; Frank and Todeschini [1994]; Danzer et al. [2001]). Neurons are arranged in layers of which modern ANNs contain at least three: The input layer, their neurons correspond to the prediction variables (the analytical values xi); input layers are counted to be layer 0 A hidden layer (sometimes more than one), representing the model of calculation and having not connection to the outside world The output layer, producing the output quantities (the measured values yj ) 166 6 Calibration in Analytical Chemistry

Fig. 6.18. Schematic representation of a multilayer perceptron with two input neurons, three hidden neurons (with sigmoid transfer functions), and two output neurons (with sigmoid transfer functions, too)

Fig. 6.19. Components and connection of two artiˇcial neurons, according to Danzer et al. [2001]

In general, there is no connection between neurons within the same layer, but each neuron is connected with each neuron of the next layer. The structure of an ANN can be seen from Fig. 6.18. Neural networks are characterized by their weights, wij, and their respective sums are given by the weight matrixes, W(j), between the diverse layers. The weights represent the strength of the directed connection between neurons i and j; see Fig. 6.19. Neurons have one or more inputs, an output, oi,anactivation state, Ai, an activation function, fact, and an output function, fout. The propagation function (net function) k neti(t)= OiWij (6.117) i=1 characterizes how the input of the neuron are computed from the outputs Oi of the other neurons and the respective weights Wij. The activation state Ai of a neuron i at time (or iteration step) t +1is given by 6.5 Calibration by Artiˇcial Neural Networks 167 ! Ai(t +1)=fact Ai(t); neti(t);i (6.118) with fact being the activation function (transfer function), neti(t) the input and i the threshold value (bias) of the neuron. This threshold must be exceeded so that the neuron becomes active (\ˇres"). The output Oi(t) of the neuron is estimated by means of the output function f out ! Oi(t)=fout Ai(t) (6.119) Often the identity function is chosen as output function. Then the output becomes ! Oi = fact neti(t) (6.120) ANNs need supervised learning schemes and can so be applied for both classiˇcation and calibration. Because ANNs are nonlinear and model-free approaches, they are of special interest in calibration. By means of the learning algorithm the parameters of the ANN are altered in such a way that the net inputs produce adequate net outputs. Mostly this is affected only by changing the weights (other procedures like adding or removing of neurons and modiˇcation of activation functions are rarely used; see Zell [1994]). The most popular techniques of multilayer perceptrons (MLP) are backpropagation networks (Wythoff [1993]; Jagemann [1998]). The weight matrixes W are estimated by minimizing the net error n 1 E = (Y O )2 (6.121) 2 i i i=1 on the basis of the generalized delta rule. According to this algorithm also the weights of inner neurons can be modiˇed. The weights are estimated from the net outputs Oi by means of Eqs. (6.117) and (6.120). Then the error function of the neurons can be calculated according to ⎧ 0 ⎨⎪ fact netij(Yij Oij)ifj is an output neuron

ı = nl (6.122) ij ⎪ 0 ⎩ fact netij ıikWjk if j is an hidden neuron k=1 0 where fact is the ˇrst derivation of the activation function, Yij the jth element of the teaching input and nl the number of neurons in the respective hidden layer (k is indexing the neurons in the next layer). The correction of the weights in the nth run is carried out by

Wjk(n +1)=ıij Oik + ˛Wjk(n) (6.123) where is the learning rate (step size parameter) and ˛ the momentum term. In case of MLP, the activation function has to be monotonous and dif- ferentiable (because of Eq. 6.122). Frequently used is the sigmoid function 168 6 Calibration in Analytical Chemistry

Fig. 6.20. Sigmoid activation function with diverse values of the steepness parameter

1 fact = Oj = (6.124) 1 + exp( netj ) with being the steepness parameter of the activation function; see Fig. 6.20. The outputs are between 0 and 1. Although for each neuron a speciˇc activation function can be deˇned, the most applications use the same function for the neurons of a given layer. When the activation function has a limited range, then X and Y must be scaled correspondingly. The threshold values of the neuron are considered by using an additional neuron in the hidden layer (\on neuron", bias neuron) the output of which is always +1. The popular radial basis function nets (RBF nets) model nonlinear relationships by linear combinations of basis functions (Zell [1994]; Jagemann [1998]; Zupan and Gasteiger [1993]). Functions are called to be radial when their values, starting from a central point, monotonously ascend or descend such as the Cauchy function or the modiˇed Gauss function at Eq. (6.125): (1) 2 (Xi Wj ) h (x ) = exp (6.125) j i r 2 where W(1) is the vector of the centres of the basis functions and r a scaling parameter. Figure 6.21 shows the dependence of the Gaussian basis function on the scaling factor. The local character of activation gets lost in case of high values of r . The determination of the weights W(2) of the output layer is simple in so far as they base on a linear model and can be calculated in one step, in 6.5 Calibration by Artiˇcial Neural Networks 169

Fig. 6.21. Gaussian radial basis function with diverse values of the scaling parameter r

Fig. 6.22. Structure of a RBF net with one hidden layer with bias neuron and one output unit as used for single component calibration (according to Fischbacher et al. [1997]; Jagemann [1998])

contrast to backpropagation nets in which the weights must be estimated iteratively. The architecture of a RBF net is given in Fig. 6.22. At ˇrst, the input of the hidden layer is calculated from the n m net input X and the k m weight matrix W(1) according to k (1) (1) 2 neti = (Xi Wj ) (6.126) j=1

With Eq. (6.126) and a Gaussian activation function the output of the hidden neurons (the RBF design matrix) becomes 170 6 Calibration in Analytical Chemistry

Fig. 6.23. A Glucose proˇle of a single test person (comparable to that of Fig. 6.13), cgluc reference value of glucose concentration, ı, and the predicted glucose values of NIR measurements, , evaluated by PLS. B Recovery function and ˇtted calibration line of the PLS calibration (number of included wavelengths n =56; 6 factors). C Recovery function and ˇtted calibration line of the RBF calibration (number of included wavelengths n = 56; 10 hidden layers), according to Fischbacher et al. [1995]

2 net(1) O(1) = exp (6.127) r

Neural networks are applied in analytical chemistry in many and diverse ways. Used in calibration, ANNs have especially advantages in case of nonlinear relationships, multicomponent systems and single component analysis in case of various disturbances. The latter applies to NIR spectroscopy used for the non-invasive determination of blood glucose by means of a ˇbre-optical measuring-head (Jagemann et al. [1995]; Muller et al. [1997]; Danzer et al. [1998]). In addition to the weak overtone and combination bands resulting from glucose, strongly disturbing absorption of water, that is the main component 6.5 Calibration by Artiˇcial Neural Networks 171

Fig. 6.24. Comparison for cross-validated PLS and RBF calibration model for different test persons (Jagemann et al. [1995]) of living tissue, takes place. Furthermore, baseline effects and some time- dependent biological processes like pulse and respiration appear. Therefore, multicomponent calibration by PLS 1 (see Sects. 6.2.6 and 6.4.2) and calibration by a RBF net have been applied. The results of PLS- and RBF calibration are compared in Fig. 6.23. It can be seen that the calibration points in case of the RBF net deviate slightly lower than that of PLS. This fact can be expressed quantitatively by means of the RMSP value (root mean standard error of prediction): n 2 (x^i xi) RMSP = i=1 (6.128) n that can be estimated by using leaving-one-out cross validation. Concerning the calibration represented in Fig. 6.23, PLS is characterized by RMSP = 1:8 mmol/L and RBF by RMSP =1:4 mmol/L. More generally, the comparison is represented in Fig. 6.24 where the RMSP values of 14 calibration models (n = 349 test persons) are given. In most cases the RBF models greatly surpasses the PLS models. 172 References References

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Danzer K, Wienke D, Wagner M (1991) Application of chemometric principles for evaluation of emission spectrographic data. Acta Cim Hungar 128:623 Deming SN, Morgan SL (1993) Experimental design: a chemometric approach, 2nd edn. Elsevier, Amsterdam Dillon WR, Goldstein M (1984) Multivariate analysis { methods and applications. Wiley, New York Draper NR, Smith H (1981) Applied regression techniques, 2nd edn. Wiley, New York Edgeworth FY (1887) On observations relating to several quantities. Hermathena 6:279 Eckschlager K, Danzer K (1994) Information theory in analytical chemistry. Wiley, New York Efron B (1982) The jackknife, the bootstrap and other resampling techniques. Society for Industrial and Applied Mathematics, Philadelphia, PA Efron B, Tibshirani R (1993) An introduction to the bootstrap. Chapman and Hall, London Ehrlich G, Danzer K (2006) Nachweisvermogen von Analysenverfahren { Objektive Be- wertung und Ergebnisinterpretation. Springer, Berlin Heidelberg New York Faber K, Kowalski BR (1997a) Improved prediction error estimates for multivariate calibration by correcting for the measurement error in the reference values. Appl Spectrosc 51:660 Faber K, Kowalski BR (1997b) Propagation of measurement errors for the validation of predictions obtained by principal component regression and partial least squares.J Chemom 11:181 Fischbacher C, Jagemann K-U, Danzer K, Muller UA, Mertes B (1995) Non-invasive blood glucose monitoring by chemometrical evaluation of NIR-spectra. Twenty-fourth EAS (Eastern Analytical Symposium), Somerset, NJ, 12-16.11.1995 Fischbacher C, Jagemann K-U, Danzer K, Muller UA, Papenkordt L, Schuler J (1997) Enhancing calibration models for non-invasive near-infrared spectroscopical blood glucose determination. Fresenius J Anal Chem 359:78 Flury B (1988) Common principal components and related multivariate models. Wiley, New York Frank IE, Todeschini R (1994) The data analysis handbook. Elsevier, Amsterdam Garden JS, Mitchell DG, Mills WH (1980) Nonconstant variances regression techniques for calibration-curve-based analysis. Anal Chem 52:2310 Hampel FR (1980) Robuste Schatzungen: Ein anwendungsorientierter Uberblick. Biometr J 22:3 Hampel FR, Ronchetti EM, Rousseeuw PJ, Stahel W (1986) Robust statistics. The approach based on inuence functions. Wiley, New York Hasselbarth W (1995) Traceability of measurement and calibration in chemical analysis. Fresenius J Anal Chem 352:400 Hskuldsson A (1988) PLS regression methods. J Chemom 2:211 Huber PJ (1981) Robust statistics. Wiley, New York 174 References

Hulanicki A (1995) IUPAC, Analytical Chemistry Division, Commission on General As- pects of Analytical Chemistry: Absolute methods in analytical chemistry. Pure Appl Chem 67:1905 Inczedy J, Lengyiel JT, Ure AM, Geleneser A, Hulanicki A (eds) (1997) Compendium of analytical nomenclature, 3rd edn (IUPAC Orange Book). Blackwell, Oxford ISO (1993), International Organization for Standardization, Guide to the expression of uncertainty in measurement. Geneva IUPAC Orange Book (1997, 2000) { printed version: Compendium of analytical nomenclature (Deˇnitive Rules 1997); see Inczedy et al (1997) { web version (as from 2000): www.iupac.org/publications/analytical compendium/ Jagemann K-U, Fischbacher C, Danzer K, Muller UA, Mertes B (1995) Application of near-infrared spectroscopy for non-invasive determination of blood/tissue glucose using neural networks. Z Physikal Chem 191:179 Jagemann K-U (1998) Neuronale Netze in der Analytik, in: Gunzler H, Bahadir AM, Danzer K, Engewald W, Fresenius W, Galensa R, Huber W, Linscheid M, Schwedt G, Tolg G, Analytiker-Taschenbuch 19:75 Kaiser H (1972) Zur Deˇnition von Selektivitat, Speziˇtat und Empˇndlichkeit von Ana- lysenverfahren. Fresenius Z Anal Chem 260:252 Lorber A, Kowalski BR (1988) Estimation of prediction error for multivariate calibration. J Chemometrics 2:93 Malinowski ER, Howery DG (1980) Factor analysis in chemistry. Wiley, New York Mandel J (1964) The statistical analysis of experimental data. Wiley, New York Mandel J (1984) Fitting straight lines when both variables are subject to error. J Quality Technol 16:1 Manne R (1987) Analysis of two partial-least-squares algorithms for multivariate calibration. Chemom Intell Lab Syst 2:187 Martens H, Ns T (1989) Multivariate calibration. Wiley, New York Moore EH (1920) On the reciprocal of the general algebraic matrix. Bull Amer Math Soc 26:394 Morgan E (1991) Chemometrics: experimental design. Wiley, Chichester, UK Muller UA, Mertes B, Fischbacher C, Jagemann K-U, Danzer K (1997) Non-invasive blood glucose monitoring by means of near infrared spectroscopy: methods for Improving the reliability of the calibration models. Int J Artiˇc Organs 20:285 Nimmerfall G, Schron W (2001) Direct solid sample analysis of geological samples with SS-GF-AAS and use of 3D calibration. Fresenius J Anal Chem 370:760 Otto M, Bandemer H (1986) Calibration with imprecise signals and concentrations based on fuzzy theory. Chemom Intell Lab Syst 1:71 Penrose R (1955) A generalized inverse for matrices. Proc Cambridge Phil Soc 51:406 Rousseeuw PJ, Leroy AM (1987) Robust regression and outlier detection. Wiley, New York Sachs L (1992) Angewandte Statistik. Springer, Berlin Heidelberg New York Sharaf MA, Illman DL, Kowalski BR (1986) Chemometrics. Wiley, New York References 175

Statistica for Windows 4.5 (1993) Statististical Software Inc Venth K, Danzer K, Kudermann G, Blaufu K-H (1996) Multisignal evaluation in IPC-MS. Determination of trace elements in Mo-Zr-alloys. Fresenius J Anal Chem 354:811 Vessman J, Stefan RI, van Staden JF, Danzer K, Lindner W, Burns DT, Fajgelj A, Muller H (2001) IUPAC, Analytical Chemistry Division, Commission on General Aspects of Analytical Chemistry: Selectivity in analytical chemistry (IUPAC recommendations 2001). Pure Appl Chem 73:1381 Wald A (1940) The ˇtting of straight lines if both variables are subject to error. Ann Math Statist 11:284 Wegscheider W (1996) Validation of analytical methods. In: Gunzler H (ed) Accreditation and quality assurance in analytical chemistry. Springer, Berlin Heidelberg New York Wold S (1974) Spline functions in data analysis. Technometrics 16:1 Wythoff BJ (1993) Backpropagation neural networks. A tutorial. Chemom Intell Lab Syst 18:115 Zell A (1994) Simulation neuronaler Netze. Addison-Wesley, Bonn Zupan J, Gasteiger J (1991) Neural networks: a new method for solving chemical problems or just a passing phase? Anal Chim Acta 248:1 Zupan J, Gasteiger J (1993) Neural networks for chemists. VCH, Weinheim 7 Analytical Performance Characteristics

As a measuring science, analytical chemistry has to guarantee the quality of its results. Each kind of measurement is objectively affected by uncertainties which can be composed of random scattering and systematic deviations. Therefore, the measured results have to be characterized with regard to their quality, namely both the precision and accuracy and { if relevant { their information content (see Sect. 9.1). Also analytical procedures need characteristics that express their potential power regarding precision, accuracy, sensitivity, selectivity, speciˇcity, robustness, and detection limit. Notwithstanding that a large number of deˇnitions exist that have been produced and harmonized by many commissions of international organizations (e.g., IUPAC, IUPAP, ISO, BIPM, IEC, IFCC, OIML), there are still ofˇcial recommendations and deˇnitions which are not completely satisfy- ing and sometimes confusing (Prichard et al. [2001]). In general, it is well-known that analytical results can be assessed and how it is done in detail. On the other hand, an assessment of analytical operations, carried out within the hierarchy of analytical techniques, methods, procedures, and SOPs, takes place scarcely according to standardized viewpoints. Figure 7.1 shows how the terms of analytical proceedings (see Chap. 9) can be classiˇed as belonging to the steps of the analytical process which is differently represented here in comparison to Fig. 2.1. The classiˇcation is descended from Gottschalk [1975], Danzer et al. [1987], Taylor [1983], and Prichard et al. [2001]. Table 7.1 illustrates by examples how different the degree of concretization of analytical proceedings is. 178 7 Analytical Performance Characteristics

Problem Object Sampling Pre-treatment Standard Operating Procedure (SOP, see Excitation ISO 78-2 [1999]). Complete Analytical Analytical Analytical Analytical Analytical Procedure, according interactions Technique Method Procedure to KAISER [1965] Detection Evaluation Validation

Fig. 7.1. The meaning of the terms of analytical proceedings in relation to the steps of the analytical process

Table 7.1. Examples of the various degree of concretization of analytical activities

Ex Analytical Analytical Analytical procedure Standard operating procedure # technique method 1 OES ICP-OES Direct ICP-OES of Complete analytical procedure wines see Thiel and Danzer [1997] 2 GC GC-MS SPME-CGC of wines Complete analytical procedure as described by De la Calle Garca et al. [1998] 3 IR NIR Non-invasive NIR Complete analytical procedure of blood-glucose see Danzer et al. [1998]

7.1 Reliability of Analytical Measurements

For the characterization of the reliability of analytical measurements the terms precision, accuracy, and trueness have a deˇnite meaning.

7.1.1 Precision

Precision is deˇned as \the closeness of agreement between independent test results obtained under stipulated conditions"(Fleming et al. [1996b]; Prichard et al. [2001]). Precision characterizes the random component of the measurement error and, therefore, it does not relate to the true value. 7.1 Reliability of Analytical Measurements 179 According to the conditions under which the measurements are carried out, it is to distinguish between

{ Repeatability, the \precision under repeatability conditions" (ISO 3534-1 [1993]) and { Reproducibility, the \precision under reproducibility conditions" (ISO 3534-1 [1993]).

Repeatability is \the closeness of the agreement between the results of successive measurements of the same measurand carried out under the same conditions of measurement". Repeatability conditions include: the same measurement procedure, the same observer, the same measuring instrument, used under the same conditions, the same location, and repetition over a short period of time (ISO 3534-1 [1993]). On the other hand, reproducibility is the \closeness of the agreement between the results of measurements of the same measurand carried out under changed conditions of measurement". The changed conditions include: principle of measurement, method of measurement, observer, measuring instrument, reference standards, location, conditions of use, and time. Such variable conditions are typical for interlaboratory studies (laboratory intercomparisons). It should be clearly distinguished whether given terms characterize analytical results or methods. In this respect, there exist some contradictory views, as can be illustrated by the example of repeatability. According to ISO 3534-1 [1993] and Prichard et al. [2001] repeatability characterizes \results of measurements", on the other hand, according to Fleming et al. [1996a] repeatability \ is a characteristic of a method not of a result". Such different interpretation of terms can occasionally be observed.

Quantiˇcation: Precision can be quantiˇed by suitable dispersion characteristics. It is proposed to characterize precision by standard deviation, see Eqs. (4.12){(4.14) and relative standard deviation, see Eq. (4.15) (Fleming et al. [1996b]; Prichard et al. [2001]). Because of some uncertainty, the characterization of analytical proceedings in their hierarchy (see Fig. 7.1) and of analytical results, respectively, will be considered in detail.

Precision of analytical techniques should only be described verbally or comparatively (e.g., \the precision of coulometry is high" or \the precision of spectrophotometry is better then that of OES"). Precision of analytical methods is usually characterized semi-quantitatively by giving a rough value of the relative standard deviation (e.g., \the precision of arc-excited OES is about 20%"). Precision of an analytical procedure is commonly expressed by an average relative standard deviation (e.g., \the precision of the determination of Mn in steel by XRF in a given routine control is 1.5%"). 180 7 Analytical Performance Characteristics

Precision of a complete analytical procedure, i.e., a standard operation procedure (SOP), should be characterized by the uncertainty of measurement (absolute or relative) as exactly as validation it stipulates. Precision of a result of measurement (y ) is characterized by the conˇdence interval of the result (see Sect. 4.1.2, Fig. 4.5 and Eqs. (4.16) and (4.17)) or the corresponding uncertainty interval (see Sect. 4.2 and Eqs. (4.29){ (4.32)). Precision of an analytical result (x) is expressed by the prediction interval which include not only the dispersion of the measured results but additionally the uncertainty of the calibration on which the estimation of x is based; see Sect. 6.2.2. Precision of a measuring system and with it also of signals and signal functions obtained by instrumental analytical methods, is characterized by the signal-to-noise ratio. In analogy to other measuring sciences, also in analytical chemistry the signal-to-noise ratio S y 1 = = (7.1) N sy syrel is used. The mean of y is mostly applied in the form of the background- corrected net value ynet S y y y y y = net = 0 = 0 (7.2) N synet synet synet where y0 is the background signal (blank) and sy0 its standard deviation; see Fig. 7.2.

y N 0 pp

Fig. 7.2. Signal function with noise; Npp noise amplitude t (peak-to-peak noise) 7.1 Reliability of Analytical Measurements 181

The net value y y 0 may be also estimated by averaging of the individual differences y y0. In both cases using differences, the belonging standard deviation synet increases in comparison with sy " 2 2 (ny 1) sy +(ny0 1) sy0 ny + ny0 synet = (7.3) ny + ny0 2 ny ny0 provided that the standard deviations sy and sy0 of the analyte signal and the background signal do not differ signiˇcantly from each other. In the case that both are estimated by an equal number of measurements (ny = ny = n) and sy sy , Eq. (7.3) is simpliˇed as follows: 0 $ 0 2 s = s (7.4) ynet y n Noise N (t) is characterized by the following characteristics:

Time average (temporal mean), N (t), where ideally is N (t) = 0, in case

that there exist a background, N (t)=y 0 Noise amplitude (peak-to-peak noise), Npp = N (t)max N (t)min;ifthe difference cannot be estimated directly, an approximation can be obtained by Npp 2y 0 (with P =0:999) 2 2 Noise variance (squared mean), N = N (t) Standard deviation of noise, N , which be roughly estimated by means of the peak-to-peak distance according to

Npp s (7.5a) N 5 (Doerffel et al. [1990]) and more exactly by

Npp s (7.5b) N where can be found in tables (Sachs [1992]; Lieck [1998]). Signal-to-noise ratio characterizes recorded signals and signal functions with regard of their quality, i.e., their precision. Unfortunately, the signal- to-noise is not uniformly used in analytical chemistry. In addition to the deˇnitions given in Eqs. (7.1) and (7.2), there exist another one, related to the peak-to-peak noise Npp: y y y y y snr(y )= net = 0 0 (7.6) Npp Npp 2(y 0)0:999 The signal-to-noise ratio snr(y ) deˇned in this way, can differ & from! S=N de- n ˇned by Eq. (7.2) considerably, namely % according to S=N 5 2 snr(y )= p ! 1 2 0;p2828 3; 536 n snr(y ) and snr(y ) 5 n S=N = n S=N, respectively. 182 7 Analytical Performance Characteristics Therefore, it is necessary to inform on the type of deˇnition which is applied, snr(y )orS=N, to avoid confusion. The signal-to-noise ratio has been used in analytical chemistry since the 1960s. At ˇrst, atomic spectroscopy prepared the way for application, and some other spectroscopic disciplines and chromatography are important domains of use.

7.1.2 Precision of Trace Analyses

It is a well-known fact that the precision in trace analysis decreases with diminishing concentration in a similar way as it does with decreasing sample weight (Sect. 2.1). The dependency of the repeatability and reproducibility standard deviation on the concentration of analytes has been investigated systematically at ˇrst by Horwitz et al. [1980] on the basis of thousands of pieces of interlaboratory data (mostly from food analysis). The result of the study has been represented in form of the well-known \Horwitz trumpet" which is represented in Fig. 7.3. The Horwitz relationship agrees with the experience of analysts and has been conˇrmed in various ˇelds of trace analysis, not only in its qualitative form but also quantitatively. Thompson et al. [2004] have estimated the mathematical form of the Horwitz functiontextscHorwitz function be- 0:85 ing sH =0:02 x , or linearized, log sH =0:85 log x. The agreement of this equation is usually good and, therefore, the Horwitz functiontextscHorwitz function is sometimes used as a bench-mark for the performance of analytical methods. For this purpose, the so-called \Horrat" (Horwitz ratio) has been deˇned, Horrat = sactual=sH , by which the actual standard deviation is compared with the estimate of the Horwitz function. Serious deviations

Fig. 7.3. The so-called \Horwitz trumpet": De- pendency of the relative standard deviation, rsd(x) on the concentration (x) 7.1 Reliability of Analytical Measurements 183 of the Horrat from 1 can be used to reject results from interlaboratory comparisons (Thompson et al. [2004]). In general, in trace analysis concessions must be made with regard to precision and accuracy. However, accuracy has strictly to be distinguished from precision.

7.1.3 Accuracy and Trueness

Accuracy is deˇned in the name of BIPM, IEC, IFCC, ISO, IUPAC, IUPAP, and OIML as \the closeness of agreement between the result of a measurement and the true value of the measurand" (ISO 3534-1 [1993]; Fleming et al. [1996b]). Accuracy characterizes the absence of a relevant bias of a measured value. As can be seen from Fig. 4.1, accuracy is a measure that combines the effects of both random and systematic deviations. Therefore, a bias can only be detected if it exceeds the range of random error. In contrast to accuracy which relates to single test results, trueness is de- ˇned as \closeness of agreement between the average value obtained from a large series of test results and an accepted reference value" (IUPAC Orange Book [1997, 2000]; Fleming et al. [1996c]). Both terms, conventional true value and accepted reference value describe practical approaches to a true value which is by nature indeterminable (ISO 3534-1 [1993]). Conventional true value is deˇned as a \value attributed to a particular quantity and accepted, sometimes by convention, as having an uncertainty appropriate for a given purpose" (ISO 3534-1 [1993]; Prichard et al. [2001]). Accuracy is a qualitative concept (ISO 3534-1 [1993]). However, in analytical practice, accuracy is frequently characterized in a quantitative way by the bias (Fleming et al. [1996b]); see Eq.(4.1).

The ISO recommendation [1993] should be followed and accuracy used only as a qualitative term. In case of quantitative characterization (by means of the bias), a problem may appear which is similar to that of precision, namely that a quality criterion is quantiˇed by a measure that has a reverse attribute regarding the property which have to be characterized. If the basic idea of measures can be accepted, which is that a high quality becomes a high value and vice versa, bias is an unsuited measure of accuracy (and trueness). In this sense, accuracy could be deˇned by means of a measure proposed in the next paragraph.

7.1.4 Remark on the Quantiˇcation of Precision, Accuracy and Trueness

Precision, accuracy and trueness are important performance characteristics in analytical chemistry. Each of them is well-deˇned in a positive sense (\closeness of agreement . . . "). However, their quantifying is done by means of unfavourable measures, namely by error quantities like, e.g., standard deviation and bias, respectively, which indeed do quantify imprecision and 184 7 Analytical Performance Characteristics inaccuracy. This inconsistency has been stated by Kaiser and Specker [1956] who criticized that \It is unsightly that the conˇdence interval is narrow when the conˇdence is high and vice versa". Therefore, they proposed an alternative measure to characterize properly the precision of an analytical measurement, namely in the form of the reciprocal value of the relative standard deviation x 1 = = (7.7) s srel It is difˇcult to comprehend why this measure has not been applied in analytical chemistry. Instead of this, in the last decades the signal-to-noise ratio has increasingly been used. Signal-to-noise ratio, see Eq. (7.1), is the measure that corresponds to in the signal domain. In principle, quantities like S=N (Eq. (7.1)) and (Eq. (7.7)) could represent measures of precision, but they have an unfavourable range of deˇnition, namely range[ ]=range[S=N]=0:::1. Useful measures of precision could be derived from relative dispersion measures, namely by their differences from 1, e.g., the precision of an analytical procedure s prec(x)=1 rsd(x)=1 x (7.8) x The numerical value of this measure increases with decreasing error. The range is normally range[prec(x)]=1::: 0, i.e. for dispersion-free measurements (sx ! 0) the precision becomes 1. On the other hand, if the dispersion amounts to 100% (sx ! x) the precision becomes 0. Therefore, high precision is characterized by a high value of prec(x). The precision becomes negative if the error exceeds the measured value what corresponds to rsd(x) > 100%. But such cases should appear very rarely in analytical chemistry. The precision of an analytical result can be quantiˇed by means of the relative conˇdence interval x prec(x)=1 (7.9) x The interpretation is similar to that of the precision of analytical procedures. Accuracy and trueness have been deˇned above and it was mentioned that these terms base on qualitative concepts (ISO 3534-1 [1993]). If it is necessary to have quantitative information, the bias, which is a measure of inaccuracy, should not be used to quantify accuracy and trueness, respectively. Instead of this, the following measures might be applied bias(x) acc(x)=1 (7.10) x and bias(x) acc(x)=1 = trn(x) : (7.11) x 7.2 Sensitivity 185

Table 7.2. Analytical results of trace elements in blood plasma analyses (Streck [2004])

Reference values/ Analytical results Recovery Precision Trueness Comment: Analyte µmol/L x ˙ x=(µmol/L) prec(x) acc(x) The result is Ca 2200 2170 ˙ 170 0.986 0.922 0.824 Correct Cu 20.0 19:5 ˙ 0:4 0.975 0.979 {0.250 Incorrecta Zn 23.0 22:8 ˙ 0:9 0.991 0.961 0.778 Correct a The reference value is outside the (upper) conˇdence limit and therefore the result is classiˇed to be incorrect. This can also be proved by the t-test

These expressions are characterized by a deˇnition range range[acc(x)] = 1 ::: 1. The following cases can appear: (i) If the range of accuracy (trueness) is 1 acc(x) > 0, then the measured result is conventionally correct (see Fig. 4.1). The more the systematic error decreases the more acc(x) approaches to 1. On the other hand, acc(x) approaches to 0 with increasing bias. (ii) If the range is acc(x) < 0, then the measured result is conventionally incorrect. The evidence of that can be seen in Fig. 4.1 and shown by means of the t-test, see Eq. (4.45). The advantage of the measures as suggested here is that they point into the same direction as it is done by the verbal deˇnitions. High precision and a high degree of accuracy, respectively, are characterized by high numerical values of the measures which approximate to 1 in the ideal case (absence of random and systematic deviations, respectively) and approximate to 0 if the deviations approach to 100%. In the worst cases, the numerical values fprec(x)g and facc(x)g become negative which indicates that the relative random or systematic error exceeds 100%. The example given in Table 7.2 is taken from a study to verify the trueness of clinical analyses (Streck [2004]). Recovery rates have been used as the criterion to accept a good agreement between the measured results and the reference values as it is frequently done by analysts. The evaluation by means of the performance characteristics prec(x) and acc(x) shows that they are meaningful measures which can usefully be applied in the daily analytical practice. From the example can also be seen that the recovery rate { solely used { can mislead the analyst if the relation of bias to conˇdence interval is disregarded.

7.2 Sensitivity

Sensitivity is a signiˇcant characteristic in all scientiˇc disciplines which have to do with measurements. Sensitivity is deˇned from the viewpoint of instrumental measuring as \the change in the response of a measuring instrument divided by the corresponding change in the stimulus" (ISO 186 7 Analytical Performance Characteristics

Fig. 7.4. Sensitivity in the case of a calibration straight line (a) and a curved calibration function (b)

3534-1 [1993]; IUPAC Orange Book [1997, 2000]; Fleming et al. [1997b], Prichard et al. [2001]). In analytical chemistry, the sensitivity SAA of an analytical procedure (of the determination of an analyte A) is deˇned as the change in the measured value divided by the corresponding analytical value (analyte amount or concentration, respectively):

dyA SAA = (7.12) dxA In case of a linear calibration function (see Chap. 6), the sensitivity becomes SAA = yA=xA and corresponds to the slope b of the calibration straight line; see Fig. 7.4a. If the calibration function is a curved line, then the sensitivity will vary according to the analyte amount or concentration as Fig. 7.4b shows.

It should be noted that the term \sensitivity" sometimes may alternatively be used, namely in analytical chemistry and other disciplines. Frequently the term \sensitivity" is associated with detection limit or detection capability. This and other misuses are not recommended by IUPAC (Orange Book [1997, 2000]). In clinical chemistry and medicine another matter is denoted by \sensitivity", namely \the ability of a method to detect truly positive samples as positive"(O'Rangers and Condon [2000], cited according to Trullols et al. [2004]). However, this seems to be more a problem of trueness than of sensitivity. It is regrettable that a uniˇed use of such an essential term like sensitivity could not be reached until now. All the more so, as the term \sensitivity" is deˇned in metrology and analytical chemistry in the same sense as it is used in daily life, namely as \reactio per actio" (effect/cause, output/input). A person is called to be sensitive if it strongly reacts on a given impulse (accusation, nudge, or stroke of fate, respectively), and insensitive vice versa. The stock market reacts sensitive (or insensitive) on changes of political and economical facts. 7.2 Sensitivity 187

Fig. 7.5. Enhancement of signal intensity by merging two single signals into a latent signal

Some analytical methods open possibilities to evaluate more than only one signal (e.g., OES, MS, XRF). In such cases, the relevant signals yA1 ;yA2 ;::: (the evaluation of each of which corresponds to another analytical procedure, see Fig. 7.1), do have different sensitivities SAA1 ;SAA2 ;:::, from which the best will be selected, as a rule. On the other hand, it is possible to evaluate simultaneously more then one signal per analyte (Danzer and Wagner [1993]) and enhance so the sensitivity of the determination of a given analyte A. The principle can be seen in Fig. 7.5 in which the intensities of two signals yA1 and yA2 are represented towards each other. The intensity of the combined signal, which is in fact a latent signal, is given by % y = y = y 2 + y 2 (7.13) Alatent A1+2 A1 A2

When merging more than two signals, yAlatent is obtained either by the n- dimensional principle of Pythagoras or { better { by principal component analysis (PCA, eigenanalysis; see Sect. 6.3; Danzer [1990]).

Because all the intensities in Fig. 7.5, yA1 , yA2 , and yAlatent , are caused by the same analyte concentration, the sensitivity also increases according to % S = S = S2 + S2 (7.14) Alatent A1+2 A1 A2

By multisignal evaluation, detection capability can be improved as has been shown for OES and MS by Danzer and Venth [1994] and Venth et al. [1996]. In multicomponent systems, the number of signals is (at least) equal in size to the number of components: 188 7 Analytical Performance Characteristics

yA = yA0 + SAAxA + SABxB + + SAN xN + eA yB = yB0 + SBAxA + SBBxB + + SBN xN + eB . (7.15) . yN = yN 0 + SNAxA + SNBxB + + SNNxN + eN

The sensitivity of each component is given by SAA;SBB;:::;SNN, analogous to Eq. (7.12). The interferences which appear between the species can be characterized by cross sensitivities as deˇned in Eq. (3.11):

@yA SAi = (7.16) @xi More extensive, multicomponent system are described by the sensitivity matrix (matrix of partial sensitivities according to Kaiser [1972], also called K-matrix according to Jochum et al. [1981]): ⎛ ⎞ SAA SAB SAN ⎜ ⎟ ⎜ SBA SBB SBN ⎟ S = ⎝ . . . ⎠ (7.17) . . . SNA SNB SNN In the ideal case only the diagonal elements of the sensitivity matrix are different from zero. Then no component disturbs any other and the analytical procedure works selectively (see Sect. 7.3). The K-matrix is deˇned analogously except that the elements are called kIJ, their deˇnition is the same as the SIJ according to Eq. (7.16). It may be useful to characterize multicomponent analyses not only by the single sensitivities of each component, but additionally by a total multicomponent sensitivity Stotal. In the case of undisturbed measurements it is given by N Stotal = Sii (7.18) i=A If interferences and overlappings appear, the total (multicomponent) sensitivity is calculated as the determinant of the (squared) sensitivity matrix (Eq. (7.17)); (Kaiser [1972]; Sharaf et al. [1986]; Bergmann et al. [1987]; Massart et al. [1988]):

Stotal = jdet(S)j (7.19) In case of serious overlappings, multivariate techniques (see Sect. 6.4) are used and p n sensors (measuring points zk) are measured for n components. From this an overdetermined systems of equations results and, therefore, non-squared sensitivity matrixes. Then the total multicomponent sensitivity is given by % T Stotal = det(S S) (7.20) (Sharaf et al. [1986]; Bergmann et al. [1987]; Massart et al. [1988]). 7.3 Selectivity and Speciˇcity 189

Some simple examples should illustrate the problem of sensitivity in multicomponent systems. At ˇrst, single component analysis of a system that contains 1 mol/L of the analyte A as well as 0.1 and 0.1 mol/L of the accompanying components B and C, i.e. 1 the concentration vector is x =(1; 0:1; 0:1); let the sensitivity be SAA = 5 au/(mol/L) and the cross sensitivities SAB =0:1 au/(mol/L) and SAC =0:2 au/(mol/L)

(i) Analysis of A in presence of the disturbing components yields the following real sensitivity according to Eq. (7.13): SAA;i =(5+0:01+0:01) au/(mol/L) = 5.03 au/(mol/L). (ii) In case that the concentrations of all the three components are equal, e.g. x =(1; 1; 1) it results for the real sensitivity: SAA;i =(5+1+2)au/(mol/L) = 8.00 au/(mol/L). (iii) In case of simultaneous multicomponent analysis, at ˇrst it should be assumed a full selective analytical procedure corresponding to the sensitivity matrix S(1) ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 500 50:10:2 512 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ S(1) = 030 ; S(2) = 0:130:1 ; S(3) = 131 : 006 0:10:26 126 The three single sensitivities can be found at the diagonal of the matrix S(1). The total multicomponent sensitivity according to Eq. (7.18) is Stotal = 90 au3/(mol/L)3. (iv) Multicomponent analysis in the case of slight disturbances as expressed by the sensitivity matrix S(2). According to Eq. (7.19) the total multicomponent sensitivity is Stotal = det(S(2))=(5 3 6+0:1 0:1 0:1+0:2 0:1 0:2) (0:230:1+50:10:2+0:10:16) = (90:0050:22)=89:785 au3/(mol/L)3.

(v) Serious disturbances as given by the sensitivity matrix S(3) results in the following total sensitivity: Stotal = det(S(3))=(5 3 6+1 1 1+2 1 2) (2 3 1+5 1 2+1 1 6) = (95 22)=73au3/(mol/L)3.

7.3 Selectivity and Speciˇcity

The concepts selectivity and speciˇcity are associated with real analytical multicomponent systems. Both selectivity and speciˇcity refer to a given state of the analytical system as well as the analytical method, i.e., a completely deˇned procedure (SOP), as characterized in Fig. 7.1. Each change in operating and physicochemical conditions like pH value, temperature as well as changed composition of the system (accompanying species, masking reagents etc) will alter the procedure and therefore cause a change of sensitivity and speciˇcity. In such a well-deˇned analytical system the term selectivity is relevant to multicomponent analysis. Selectivity of an analytical procedure characterizes the extent to which n given analytes can be measured simultaneously by n sensors (detecting channels) without interferences by other components

1 au: arbitrary unit 190 7 Analytical Performance Characteristics and, therefore, can be detected or determined independently and undisturbedly (Kaiser [1972]; Danzer [2001, 2004]). On the other hand, speciˇcity refers to single component analysis and means that the one individual analyte can be undisturbedly measured in a real sample by a speciˇc reagent, a particular sensor or a comparable measuring system (e.g., measurement of emitted or absorbed radiation at a ˇxed wavelength). Selectivity and speciˇcity are important performance characteristics of analytical procedures, especially in connection with validation processes. Nevertheless, both terms are used mostly verbal and a quantiˇcation is avoided, as a rule (IUPAC; see Vessman et al. [2001]). Moreover, the concepts of selectivity and speciˇcity are used interchangeably and synonymously. Occasionally, speciˇcity is regarded as an intensiˇcation of selectivity, viz. the ultimate of selectivity (den Boef and Hulanicki [1983]; Persson and Vessman [1998, 2001]; Prichard et al. [2001]). A clear differentiation between selectivity and speciˇcity was ˇrstly given by Kaiser [1972] together with a quantiˇcation of the both characteristics. According to him an analytical procedure is known as \selective" if it can detect and determine simultaneously several components independently from each other. On the other hand, a procedure is known as \speciˇc"ifonly one species can be detected and determined independently from all the other components which are present in the sample but does not contribute to the signal in this case. Starting from a relationship like Eq. (6.70a), Kaiser [1972] deˇned sensitivity, partial sensitivities (cross sensitivities, Eq. (7.16)), and the sensitivity matrix Eq. (7.17). From these quantities he derived the following measures:

Selectivity: ⎛ ⎞ ⎜ ⎟ ⎜ SIJ ⎟ = min ⎜ 1⎟ (7.21) I=1:::N ⎝ N ⎠ SIJ jSIIj J=1 Speciˇcity:

SAA A = 1 (7.22) N SJJ jSAAj J=1 These quantiˇcations are inconvenient in two respects:

2 (i) The range of both quantities is Rf g = RfAg =(1 :::+ 1) and therefore, an interpretation of a given value of and A in the sense that the selectivity/speciˇcity is high or less, may be difˇcult. (ii) The deˇnitions at Eqs. (7.21) and (7.22) do not consider the concentration of the disturbing components. However, not only the cross sensitivity but also the concentration of trouble-making components is crucial for the appearance and intensity of disturbing effects; see Eq. (3.10)

2 The range Rfxg of values of a quantity x corresponds to its deˇnition range 7.3 Selectivity and Speciˇcity 191

The last item has been taken into account by Fujiwara et al. [1980] who introduced a \total selectivity" similar to Kaiser's selectivity (Eq. 7.21) but products SIJ cJ instead of the partial sensitivities. However, this deˇnition range is (1 :::+ 1). A more acceptable range of \selectivity" results from Doerffel's \coefˇcient of selectivity CSA"(Doerffel et al. [1986]): y S c CS = A = AA A (7.23) A N N yI SAI cI I=A I=A (SAA sensitivity of the detection channel A for the analyte A; SIA partial sensitivities of all components I = A;B;:::;N at the detection channel A; cA concentration of the analyte A; cI concentrations of all components I = A;B;:::;N). But unfortunately, Doerffel's \selectivity" according to Eq. (7.23) proves to be speciˇcity because it corresponds to Kaiser's Eq. (4.54). Starting from the sensitivity matrix, Eq. (7.17), as it is illustrated in Fig. 7.6a, selectivity can be deˇned as follows: N SII xI sel(A;B;:::;N)= I=A (7.24) N N SIJ xJ I=A J=A

Fig. 7.6. Selectivity matrix (a) and speciˇcity vector of the detecting channel A (b) with the simpliˇed formulas (all the xI and xJ = 1) for selectivity and speciˇcity (on the right) 192 7 Analytical Performance Characteristics In the ideal case, only the diagonal elements of the matrix at Eq. (7.17) are different from zero and all the off-diagonal elements SIJ = 0 (for I =/ J), sel(A;B;:::;N) becomes 1. For real analytical systems, Eq. (7.24) expresses a certain degree of selectivity depending on the sensitivities of the analytes I, their contents xI , and all the cross sensitivities SIJ and contents xJ of potentially disturbing species. For the characterization of speciˇcity only the detecting channel of the analyte, A, is relevant and therefore only the corresponding vector

sA =(SAA;SAB;:::;SAN ) (7.25) is of interest. It contains the sensitivity SAA and the cross sensitivities SAJ at the channel (sensor, wavelength, etc.) A; see Fig. 7.6b. Speciˇcity of an analyte A with reference to the accompanying components B;C;:::;N can be deˇned by S x spec(A=B;:::;N)= AA A (7.26) N SAJ xJ J=A The problem is illustrated in a simpliˇed way in Fig. 7.6. The quantities sel(A;B;:::;N) and spec(A;B;:::;N) not only deˇne absolute characteristics but also a certain degree of selectivity and speciˇcity of an analytical procedure. The measures can take values from 1 for fully selective or totally speciˇc procedures, respectively. The range of selectivity Rfselg and the range of speciˇcity Rfspecg are given by Rfselg = Rfspecg =(0:::+ 1) (7.27) For practical purposes, it can be supposed that values of sel > 0:9 and spec > 0:9 indicate fair selectivity and speciˇcity, respectively, whereas smaller values show more and more insufˇcient selectivity and speciˇcity. To obtain some impression on the characteristics, the same examples as dealt with in Sect. 7.2 (iii){(v) will be considered in Table 7.3. Some more examples from the application of sensors, photometry, and ion sensitive electrodes have been given by Danzer [2001]. Not only components from the set A;:::;N can interfere with the analytes but also additional species which may be partly unknown or will be formed during the measuring process. Such situations occur especially in ICP-MS, where the signal of an isotope Ai may be interfered by isotopes of other elements, Bj , Ck etc., and additionally by molecule ions formed in the plasma (e.g., argon compound ions and ions formed from solvent constituents). In such cases the sensitivity matrix, Eq. (7.17), will be extended into ⎛ ⎞ SAA SAB SAN SA;N +1 SA;N +p ⎜ ⎟ ⎜ SBA SBB SBN SB;N+1 SB;N+q ⎟ S = ⎝ . . . . . ⎠ (7.28) . . . . . SNA SNB SNN SN;N+1 SN;N+r 7.3 Selectivity and Speciˇcity 193

Table 7.3. Selectivities and speciˇcities in case of various sensitivity matrixes

Sensitivity matrix Selectivity Speciˇcities (see Sect. 7.2 (iii){(v)) spec(A=B; C)=1:00 S(1) sel(A; B; C)=1:00 spec(B=A;C)=1:00 spec(C=A;B)=1:00 spec(A=B; C)=0:943 S(2) sel(A; B; C)=0:946 spec(B=A;C)=0:938 spec(C=A;B)=0:952 spec(A=B; C)=0:625 S(3) sel(A; B; C)=0:636 spec(B=A;C)=0:600 spec(C=A;B)=0:667

The cross sensitivities SJ;N+i can easily be estimated in the case of neighbouring isotopes, viz. from the isotope abundances. On the other hand, estimation in case of molecule ions may be difˇcult and can only be done experimentally.

The concept of selectivity and speciˇcity has been applied to characterize interferences appearing in two different ICP-MS techniques (Horn [2000]). Classical ICP-MS with pneumatic nebulization and ETV-ICP-MS are compared for the determination of traces of zinc in sea-water. Whereas spectral interferences decrease using the ETV device, nonspectral interferences increase signiˇcantly (Bjorn et al. [1998]). A quantitative comparison of the both analytical procedures, here called PN (pneumatic nebulization) and ETV (electrothermal vaporization, Sturgeon and Lam [1999]) is possible by means the speciˇcity as a function of the Zn concentration (Horn [2000]). The spectral interferences on the four zinc isotopes are listed in Table 7.4.

Table 7.4. Seawater components and their spectral interferences with zinc isotopes

Ion 64 amu 66 amu 67 amu 68 amu Concentration in CASS-3 Zn2+ n.a.a 48.6% n.a. 27.9% n.a. 4.1% n.a. 18.8% Certiˇed 1.24 ng/mL Cl {{35Cl16O16O+35Cl16O17O+ About 16 mg/L 2 32 16 16 +3416 16 +3334 +3616 16 + SO4 S O O S O O S S S O O 32S32S+32S34S+32S17O18O+34S36S+ 33S33S+33S16O18O+34S17O17O+ 32S17O17O+33S17O17O+34S16O18O+ About 2 mg/L 32S16O18O+34S16O17O+34S34S+ 33S16O17O+33S17O18O+ 32S18O18O+ Mg2+ 24Mg40Ar+26Mg40Ar+ { { About 1 mg/L 26Mg38Ar+ a Natural abundance 194 7 Analytical Performance Characteristics

The speciˇcity vectors of the four zinc isotopes are given in matrix form as follows: ⎛ ⎞ I64 I64 I64 I64 ⎜ ⎟ ⎜ xZn2+ xCl xSO2 xMg2+ ⎟ ⎜ 4 ⎟ ⎜ ⎟ ⎜ I66 I66 I66 I66 ⎟ ⎜ ⎟ ⎜ xZn2+ xCl xSO2 xMg2+ ⎟ ⎜ 4 ⎟ S = ⎜ ⎟ (7.29) ⎜ I67 I67 I67 I67 ⎟ ⎜ ⎟ ⎜ xZn2+ xCl xSO2 xMg2+ ⎟ ⎜ 4 ⎟ ⎝ I68 I68 I68 I68 ⎠

x 2+ xCl x 2 x 2+ Zn SO4 Mg The results for the PN- and ETV procedure are the following: ⎛ ⎞ 2+ 2 2+ Zn Cl SO4 Mg ⎜ ⎟ ⎜ 3 2 2 ⎟ ⎜ 64 amu 616 1:22 10 1:68 10 2:85 10 ⎟ ⎜ ⎟ ⎜ 4 4 2 ⎟ SPN = ⎜ 66 amu 378 3:48 10 6:48 10 1:43 10 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 67 amu 63 6:03 104 1:36 104 2:52 103 ⎠ 68 amu 265 1:99 104 2:15 104 8:97 102 ⎛ ⎞ 2+ 2 2+ Zn Cl SO4 Mg ⎜ ⎟ ⎜ 2 2 2 ⎟ ⎜ 64 amu 5669 1:02 10 1:30 10 6:24 10 ⎟ ⎜ ⎟ ⎜ 3 2 2 ⎟ SET V = ⎜ 66 amu 3557 5:67 10 2:30 10 5:69 10 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 67 amu 524 1:01 103 2:53 103 2:80 103 ⎠ 68 amu 2438 4:13 103 1:73 102 4:40 102 where the partial sensitivities are given in counts mL s1 ng1. The partial sensitivities are the result of both spectral and nonspectral interferences. It can be seen 2 that the partial sensitivities of the four masses due to the interfering ions Cl ,SO4 and Mg2+ are low compared to the sensitivities of Zn2+. Furthermore it can be seen that ETV-ICP-MS is about ten times more sensitive than PN-ICP-MS. It is possible to calculate so-called speciˇcity functions which describe the speciˇcity of a procedure in dependency of the concentration of the analyte, here Zn, in the sample: 2+ 2 2+ 616 xZn2+ specPN;64 amu Zn =Cl ; SO4 ; Mg = 616 xZn2+ + 93300 2+ 2 2+ 5669 xZn2+ specET V;64 amu Zn =Cl ; SO4 ; Mg = 5669 xZn2+ + 153000 The both functions are represented in Fig. 7.7. By means of the speciˇcity function, concentration-matrix ratios can be estimated for which a reliable determination of analytes may be possible.

For speciˇc applications, measures like selectivity indices, selectivity factors, and selectivity coefˇcients have been introduced. Their signiˇcance is 7.4 Robustness and Ruggedness 195

Fig. 7.7. Speciˇcity functions of the determination of Zn in seawater by means of PN-ICP-MS and ETV-ICP-MS limited to certain ˇelds of analytical chemistry like sensors, ISE, and chromatography as well as concrete applications. As expected, the concept of selectivity and speciˇcity is closely related to that of sensitivity. The same may be anticipated for the concept of robustness and ruggedness.

7.4 Robustness and Ruggedness

Methods can only usefully applied in analytical practice when they are suf- ˇciently robust and therefore insensitive to small variations in method conditions and equipment (replacement of a part), operator skill, environment (temperature, humidity), aging processes (GC- or LC columns, reagents), and sample composition. This demand makes robustness (ruggedness) to an important validation criterion that has to be proved by experimental studies. The concepts of robustness and ruggedness mostly have been described verbally where it must be stated that their use is frequently interchangeably and synonymously (e.g., Hendricks et al. [1996]; Kellner et al. [1998]; EURACHEM [1998]; ICH [1994, 1996]; Wunsch [1994]; Wildner and Wunsch [1997]; Valcarcel [2000]; Kateman and Buydens [1993]). Only in a few cases was there a distinction made between both concepts by their use in intra laboratory studies (robustness) and interlaboratory studies (ruggedness); see USP 23-NF18 [1995]; USP 24-NF19 [2000]; Ro- driguez et al. [1998]; Zeaiter et al. [2004]. Wahlich and Carr [1990] seem to be the ˇrst to use robustness and ruggedness in a hierarchical sense but in a reverse meaning as given above. ICH [1994] deˇnes robustness as follows: \The robustness of an analytical procedure is a measure of its capacity to remain unaffected by small, but deliberate variations in method parameters and provides an indication of its reliability during normal use". 196 7 Analytical Performance Characteristics USP 23-NF18 [1995] and USP 24-NF19 [2000] accept this deˇnition of robustness and deˇne ruggedness so: \The ruggedness of an analytical method is the degree of reproducibility of test results obtained by the analysis of the same samples under a variety of conditions such as different laboratories, different analysts, different instruments, different lots of reagents, different elapsed assay times, different assay temperatures, different days, etc. Rugged- ness is normally expressed as the lack of inuence on test results of operational and environmental factors of the analytical method. Ruggedness is a measure of reproducibility of test results under the variation in conditions normally expected from laboratory to laboratory and analyst to analyst." Whereas this deˇnition of ruggedness directly refers to inter-laboratory studies, the ICH deˇnition of robustness clearly is related to intra-laboratory studies. Consequently, it was proposed to deˇne (Burns et al. [2005]): \Robust- ness of an analytical procedure is the property that indicates insensitivity against changes of known operational parameters on the results of the method and hence its suitability for its deˇned purpose" and \Ruggedness of an analytical procedure is the property that indicates insensitivity against changes of known operational variables and in addition any variations (not discovered in intra-laboratory experiments) which may be revealed by inter-laboratory studies"(Burns et al. [2005]). The conditions and factors which inuence uncertainty and insensitivity of analytical procedures obtained in a given laboratory or in different laboratories can be considered in detail and described quantitatively (Danzer [2004]; Burns et al. [2005]). As a starting point the general Eq. (3.15), particularly Eq. (3.15a) will be taken, which describes the gross signal in dependence from all signal effects from the analyte as well as from blank, interferences, and inuencing factors:

yA = f (xA; xB;xC ;:::;xN ; f1;:::;fm) N m (7.30) = yA0 + SAAxA + SAixi + IAj xj + eA i=B j=1 The robustness of an analytical procedure for the determination of the analyte A in presence of some accompanying species i = B;:::;N under inuence of various factors fj (j =1;:::;m) according to Eq. (4.30) is in reciprocal proportion to the sum of all their cross sensitivities, SAi, multiplied by the actual amounts, xi, and the speciˇc inuencing strengths, IAj , of the factors multiplied by their actual values (in relation to xA); see Danzer (2004). Because of the way measurements are obtained, the range of their values is range =(0:::1), so it makes sense to calculate the relative robustness which includes the analyte sensitivity and amount itself, SAA xA, as follows:

SAAxA rob(A=B;:::;N; f1;:::;fm)= # # (7.31) N m # # SAAxA + jSAij xi + IAj xj i=B j=1 7.4 Robustness and Ruggedness 197 Relative robustness will have values between 0 (no robustness) and 1 (ideal robustness). Burns et al. [2005] deˇnes the relative robustness of an analytical procedure as \the ratio of the ideal signal for an uninuenced method compared to the signal for a method subject to known operational parameters determined in an intra-laboratory experiment"(Burns et al. [2005]). For the deˇnition of ruggedness, which refers to inter-laboratory situations, some further (unknown) effects uAk must be added to Eq. (7.30):

yA = f (xA; xB;xC ;:::;xN ; f1;:::;fm; u1 :::;up )

N m p (7.32) = yA0 + SAxA + SAixi + IAj xj + uk + eA i=B j=a k=1 and therefore, relative ruggedness is obtained according to

rug(A=B;:::;N; f1;:::;fm; u1;:::;up ) S x = AA A (7.33) N m # # p # # SAAxA + jSAij xi + IAj xj + jukj i=B j=1 k=1 The range of relative ruggedness is rangefrugg =(0:::1), too. According to Burns et al. [2005], the relative ruggedness of an analytical procedure is given \by the ratio of the ideal signal for an uninuenced method compared to the signal for a method subject to known and unknown operational parameters as determined in an inter-laboratory experiment". The problem, particularly in analytical practice is that most of the parameters needed for the calculation of relative robustness and ruggedness may frequently be unknown. In such cases it may be possible to check on robustness and ruggedness by means of statistical tests (see Sect. 4.3). All the variations to the measured signal, apart from that of the analyte, can be considered in form of error terms; see Eq. (3.6): N SAixi = ei (7.34a) i=B m IAj xj = ej (7.34b) j=1 p uk = ek (7.34c) k=1 and Eq. (7.30) turns into

yA = yA0 + SAxA + ei + ej + eA = yA0 + SAxA + eij + eA (7.35) 198 7 Analytical Performance Characteristics (see Eqs. (3.16a){(3.16c)) and Eq. (7.32) turns into

yA = yA0 + SAxA + ei + ej + ek + eA = yA0 + SAxA + eijk + eA (7.36) where the error contributions are additive: ei + ej = eij are the intra- laboratory variations and ei + ej + ek = eijk the intra-laboratory variations.

Test of robustness: On the basis of Eq. (7.35), robustness as an intra- laboratory property can be tested in three ways: (i) Inuence on precision can be tested as usual by Fisher's F -test 0 with the null hypothesis H0: total = A and therefore H0: ij =0; see Sect. 4.3.3: 2 2 2 + 2 ^ total ijA ij A F = 2 = 2 = 2 (7.37) A A A ^ If F F1˛;1;2 , then the null hypothesis cannot be rejected and the procedure may be considered as to be robust (strictly speaking robust with regard to method precision). (ii) Inuence on sensitivity can be tested by Student's t-test. The null real ideal ideal hypothesis is H0: SAA = SAA where SAA is the sensitivity under real ideal, i.e. robust conditions and SAA the sensitivity under the inuence of i interferents and j factors: # # #Sideal Sreal# t^ = AA AA (7.38) SAA t˛;

If t>t^ 1˛; then the real sensitivity differs signiˇcantly from the ideal one and a nonlinear error is proved. (iii) More in detail, each factor (interferents i and inuence factors j) can individually be tested by means of multifactorial experiments where each factor is usually varied at two levels. The planning and evaluation of such a multifactorial design is given in Sect. 5.1.2. The signiˇcance of each cross sensitivity and factor can be tested by an individual t-test: # # # # jS j IAJ t^ = AI ; t^ = (7.39) A t1˛; A t1˛;

If jSAI j exceeds the conˇdence interval At1˛; of the experimental error eA, then the inuence of the factor concerned is signiˇcant and robustness against this factor is missing. On the other hand, jSAI j <A t1˛; shows robustness against the particular interferent or factor, respectively.

Test of ruggedness: Robustness is regarded to be an inter-laboratory property. In this case, all the terms in Eq. (7.36) are relevant and ruggedness can be tested similar to robustness in the same three ways: 7.4 Robustness and Ruggedness 199 (iv) Inuence on precision: 2 2 2 + 2 + 2 ^ total ijkA ij k A F = 2 = 2 = 2 (7.40) A A A The total error has to be calculated here in different way compared with 2 2 robustness. Whereas in (i) ijA is the variance within a laboratory, ijkA is the variance between laboratories plus that within the laboratories, 2 2 2 ^ ijkA = k + ijA. The interpretation is analogous to (i), if F F1˛;1;2 , then the null hypothesis cannot be rejected and the procedure can be considered as to be rugged. Advantageously, the test can be carried out by schemes of ANOVA (analysis of variance) as given in Sect. 5.1.1. In most cases, tests analogous to (ii) and (iii) which exceed the situation of a single laboratory do not make much sense because there do not exist an overall sensitivity for all the laboratories. Also the cross sensitivities are individual for each lab, as a rule. Therefore, such individual measures will not be merged. In the following example, robustness and ruggedness of a procedure used in two laboratories, L1 and L2, will be considered. It is supposed that they use the same procedure to determine the analyte A in the presence of the interferents B and C under the inuence of the factors a, b, and c. The input data for the computation are given in the following table.

Lab Sensitivity Cross sensitivities Inuence strengths Experimental Degrees of SAA SAB SAC IAa IAb IAc error sA freedom L1 10 1.2 1.5 0.25 {0.13 0.08 1.2 10 L2 9 1.4 1.9 0.22 {0.14 0.10 1.5 10 (a) Robustness of the procedure in lab L1 where the following conditions exist: xB =0:11, xC =0:08, xa =1:5, xb =0:7, xc =1:3

robL1(A=B; C; a; b; c)

10 = 10+0:132+0:120+0:375+0:091+0:104

10 = =0:924 10+0:822

Thus the relative robustness of the lab L1 is 92.4%.

Using the F -test, ^ 2 2 F = ijA=A

0:1322 +0:122 +0:3752 +0:0912 +0:1042 +1:22 = 1:22

=1:123

(b) Robustness in lab L2 where the following conditions exist: xB =0:14, xC =0:13, xa =1:8, xb =1:7, xc =1:6

robL2(A=B; C; a; b; c)

10 = 10+0:196+0:247+0:396+0:238+0:160

10 = =0:808 10+1:237

Thus the relative robustness of the lab L2 is 80.8%. Using the F -test, ^ 2 2 F = ijA=A

0:1962 +0:2472 +0:3962 +0:2382 +0:1602 +1:52 = 1:52

=1:15 F(0:05; 10; 10) = 2:98 swithin 0:03295 and ruggedness cannot be conˇrmed. The unknown inuences, ek can be 2 2 estimated from the total variance sbetween = eijkA= and the averaged sums of eijA. The sums of eij are given only for completeness. From this the robustness of the separate procedures has been proven as shown in (a) and (b). 7.5 Limit Values 201 7.5 Limit Values

Limits characterize the detection capability of analytical methods and can be related to both analytical domains, sample domain as well as signal domain. Although there are several limits, namely lower and upper limits3 as well as thresholds, the most important problem in analytical chemistry is the distinction between real measurement values and \zero" values or blanks, respectively. Notwithstanding different mathematical derivations and experimental determination, the following three measures are necessary as well as sufˇ- cient for the characterization of analytical procedures:

(1) The critical measurement value (critical value, CV), yc, being the lowest signal which can signiˇcantly be distinguished from that of a blank sample

(2) The limit of detection (detection limit, LD), xLD , being the lowest content (concentration) which produces a signal y yc with high statistical probability (3) The limit of quantiˇcation (LQ, limit of determination, quantitation limit), xLQ, being the lowest content which can quantitatively be determined with a given precision (mostly expressed by the relative uncertainty of measurement) These limits are the fundamental measures to characterize the detection and quantiˇcation abilities of analytical procedures and make decisions on measured values and analytical results. Whereas CV and LD can be determined in a general way on the basis of objective statistical conditions, LQ can only be estimated on the basis of subjective demands resulting from a given analytical problem. In addition, there exist a lot of different terms, symbols, deˇnitions, and meaning for limits within the scientiˇc community. Some of them seem to be confusing in deˇnition and application and contradicting amongst themselves, too; see Currie [1992, 1995, 1997]; Ehrlich and Danzer [2006]. In general the limits can be derived in a simple and understandable way but care has to be taken in interpreting them. Before dealing with the limits CV, LD, and LQ, their meaningful position within the system of analytical domains will be shown (Table 7.5) Whereas the limits ACV (Kaiser's \3" limit, Kaiser and Specker [1956]), MLD, and MLQ are only of special interest within the corresponding domains and, therefore, of indirect importance for analytical problems, CV, LD, and LQ are the relevant quantities to characterize the performance of a complete analytical procedure according to Kaiser [1965] and a SOP according to ISO 78-2 [1999]. 3 Because the distinction from upper limits (100%) represents only the reverse situation to that from lower limits and is relatively infrequent, too, it will not be treated here. The mathematical models are equivalent to that used for evaluation of lower limits (Doerffel [1990], pp 97; Doerffel et al. [1990], pp 123) 202 7 Analytical Performance Characteristics

Table 7.5. Analytical limits within the signal and sample domain the most meaningful of which are emphasized

Signal domain (measured values) Sample domain (analytical values) Critical measuring value, CV Analytical value at CV (ACV) Measured value at LD (MLD) Limit of detection, LD Measured value at LQ (MLQ)(also min- Limit of quantiˇcation, LQ imum quantiˇable value, IUPAC [1995])

The statistical fundamentals of the deˇnition of CV and LD are illustrated by Fig. 7.8 showing a quasi-three-dimensional representation of the relationship between measured values and analytical values which is characterized by a calibration straight line y = a + bx and their two-sided conˇdence limits and, in addition (in z-direction) the probability density function the measured values. The following assumptions are made for the statistical treatment of the detection problem: (i) Feasibility of repeated analyses, namely of the complete analytical procedure from sampling to data evaluation, not only repetitions of instrumental measurements (ii) Normal distributions of the measured values (iii) Homogeneity of variances (homoscedasticity) (iv) Known calibration (absolutely known or estimated by calibration experiments); in the last-mentioned case the uncertainty intervals of the calibration function must be known (v) Purpose of evaluation: it must be clear if the limits are estimated to characterize the performance of an analytical procedure before of its application (a priori) or to interpret an analytical result obtained by this procedure (a posteriori)

The critical value yc represents the smallest measurement value that can be distinguished from the blank yBL with a given level of signiˇcance P =1 ˛. In the most general case, the critical value is estimated from the average blank and its uncertainty interval U (y BL)

yc = y BL + U (y BL) : (7.41a) As a rule, the average blank is estimated from repetition measurements of a { not too small { number of blank samples as arithmetic mean y BL.If there is information that another than normal distribution applies, then the mean of this other distribution should be estimated (see textbook of applied statistics; see Arnold [1990]; Davies and Goldsmith [1984]; Graf et al. [1987]; Huber [1981]; Sachs [1992]). The uncertainty of the blank (or the background noise) is estimated from the blank scattering which is characterized by corresponding error quanti- 7.5 Limit Values 203

Fig. 7.8. Schematic three-dimensional representation of a calibration straight line of the form y = a+bx with the limits of its two-sided conˇdence interval and three probability density function (pdf) p(y ) of measured values y belonging to the analytical values (contents, concentrations) x(A) = 0 (A), x = x(B) (B) and x(C) = xLD (C); yc is the critical value of the measurement quantity; a the intercept of the calibration function; y BL the blank; x(B) the analytical value belonging to the critical value yc (which corresponds approximately to Kaiser's \3-limit"); xLD limit of detection

ties, viz statistical or combined one, such as standard deviation sBL or combined standard uncertainty u(yBL); see Sect. 4.2; AMC [1995]; EURACHEM [1995]; Thompson [1995]. The extended uncertainty will be obtained by means of a coverage factor k according to Eq. (4.29). With this the critical value is given by

yc = y BL + k u(y BL) (7.41b) and in the special case of statistical estimation

yc = y BL + k sBL : (7.41c) Applying Eq. (7.41b) or Eq. (7.42c), the factor k can be chosen in various ways, namely

k = 3: This value was proposed by Kaiser [1965] and it is high enough to tolerate differences from the normal distribution; the levels of sig- niˇcance are about P 0:998 in case of normally distributed values, P 0:95 for non-normal unimodal distributions, and P 0:90 for any one distribution. k = 2: If the blanks are normally distributed with high probability, then a signiˇcance level P =0:95 is already given for a sample size of n =6.

k = f1˛; : The quantile of Student's t-distribution represents { also assuming a normal distribution { an individual factor which considers explicitly both the experimental expense ( = n 1) and the signiˇcance level P =1 ˛. 204 7 Analytical Performance Characteristics

k may also be estimated by means of special knowledge of the concrete distribution or on the basis of distribution-free estimations, e.g., resampling techniques (see Ehrlich and Danzer [2006]).

When the critical value is exceeded, y yc, then the analyte under examination is proved to be present in the sample. A quantiˇcation of the critical value by means of a calibration factor S = b or function y = a + bx is not very meaningful, but when it is done notwithstanding, then two consequences must be taken into account:

(1) The analytical value ACV (see Table 7.5) which corresponds to xACV = x(B) in Fig. 7.8 have a relative error of about 33% if the calculation is carried out according to Eq. (7.41c) with k = 3 (from xACV =3sBL=b 1 3sACV it results in sACV =xACV =3 and, therefore, sACV;rel =0:33. The relative uncertainty of xACV is xACV =xACV xACV =3sACV = 1 and amounts therefore to 100% (2) As can be seen from the distribution function B in Fig. 7.8, an analytical value xACV produces only in 50% of all cases signals y yc. Whereas the error of the ˇrst kind (classifying a blank erroneously as real measurement value) by the choice of k =2:::3 can be aimed at ˛ 0:05, the error of the second kind (classifying a real measured value erroneously as blank) amounts ˇ 0:5. Therefore, this analytical value { which sometimes, promoted by the early publications of Kaiser [1965, 1966], plays a certain role in analytical detection { do not have any signiˇcance as a reporting limit in case of y

xACV = U (y BL)=b : (7.42) In case of experimental calibration, the uncertainty of both the blank and the calibration coefˇcient, U (y BL;b), have to be consider, e.g. according to $ U (y ;b) s t 1 1 x = BL = BL 1 ˛; + (7.43) ACV b b N n with n being the number of blanks ( = n 1 are the corresponding degrees of freedom) and N being the number of repetition measurements at the analysis. Because the relation at Eq. (7.43) is unsuited as a limit in the concentration domain because of the large risk of error ˇ =0:5, a useful measure must be derived. In order to obtain a comparable risk for the error of second kind (˛ = ˇ 0:05), a deˇnition of the limit of detection has to consider conˇdence 7.5 Limit Values 205 intervals of both blanks and measurement values (Kaiser [1965, 1966]; Ehrlich [1969]; Currie [1992, 1995, 1997]): $ sCV t1ˇ; 1 1 x = x + + (7.44) LD ACV b N n Assuming that the standard deviation of the critical value is approximately equal to the standard deviation of the blank, sCV sBL, it results for ˛ = ˇ in $ s t 1 1 x =2 BL 1 ˛; + (7.45) LD b N n With this the LD is the double of the ACV (Eq. 7.43). The limit of detection can also be estimated by means of data of the calibration function, namely the intercept a which is taken as an estimate of the blank, a y BL, and the conˇdence interval of the calibration straight line: " 2 1 1 xcal xLDcal = xACVcal + sx0 t1ˇ; + + (7.46) N n Sxx with sx0 = sy:x=b being the standard deviation of the calibration procedure at the position xi =0, xcal is the average of all the calibration contents n 2 (centre of calibration data) and Sxx = (xi x) is the sum of squares of i=1 the contents of the calibration samples. The number of the degrees of freedom is = n 2 in this case. With sx0(xi =0) sx0(xi = xACV ) Eq. (7.46) becomes " ! 2 1 1 xcal xLDcal = sx0 t1˛; + t1ˇ; + + N n Sxx " 2 1 1 xcal xLDcal =2sx0 t1˛; + + 2 xACV(cal) (7.47) N n Sxx The limit of detection is that analytical value that always will be found, apart from the small risk of error ˛ = ˇ. The detection limit characterizes analytical procedures with regard to minimal value which can be detected with high signiˇcance. The limit of detection can, therefore, be reported as limit content when no signal is found. The corresponding measured value at LD (see Table 7.5) is not of crucial importance in analytical chemistry. It characterizes that signal which can signiˇcantly be distinguished from the blank considering both types of error (˛ and ˇ). Because of its relatively high uncertainty of measurement of 50%, the limit of detection is mainly used as a detection criterion that the analyte is present in the sample. Taking as the basis a coverage factor of k =6, 206 7 Analytical Performance Characteristics contents at the LD are characterized by a relative standard deviation of about 1 17% (from xLD =6sBL=b 6sLD it results sLD =xLD =6 and, therefore, sLD;rel 0:17). The relative uncertainty of xLD is xLD =xLD xLD =3sLD =0:5 and amounts therefore to 50%. For this reason, a number of analysts uses a further limit quantity, namely the limit of quantiˇcation, xLQ,(limit of determination), from which on the analyte can be determined quantitatively with a certain given precision (Kaiser [1965, 1966]; Long and Winefordner [1983]; Currie [1992, 1995, 1997]; IUPAC [1995]; Ehrlich and Danzer [2006]). This limit is not a general one like the critical value and the detection limit which are deˇned on an objective basis. In contrast, the limit of quantiˇcation is a subjective measure depending on the precision, expressed by the reciprocal uncertainty xLQ=xLQ = k, which is needed and set in advance. The limit of quantiˇcation can be estimated from blank measurements according to $ s t 1 1 x = k BL 1 ˛; + (7.48) LQ b N n or from calibration data " 2 1 1 (xLQcal xcal) xLQcal = ksx0 t1˛; + + (7.49) N n Sxx In the analytical practice, LQ bases frequently on a precision that is characterized by a relative uncertainty of 10% and, therefore, xLQ =10u(xLQ) is applied. A relative uncertainty of 5% would require xLQ =20u(xLQ), 1% requires xLQ = 100u(xLQ) and so on. The quantiˇcation limit is relevant to concentration domain. Reporting LQs should always be completed by the conditions of precision. The limits CV , LD, and LQ are widely applied in analytical chemistry but, unfortunately, not in a standardized manner. Various concepts are in use as well as different terms, symbols, deˇnitions and meanings. Overviews are given and consequences are shown among others by Currie [1995, 1997]; Long and Winefordner [1983]; Fleming et al. [1997a]; Gei and Einax [2001]; Ehrlich and Danzer [2006]. The term detection capability (detection power) represents a generic term of the performance of analytical methods at the lower limit of applicability. Mostly it is used descriptively (detection capability of a method is \high", \good" or \sufˇcient") or for giving order of magnitudes (detection capability is in the \ppm-range" or \ppb-range").

A quantiˇcation of the detection power is possible, in principle, namely in analogy to diverse deˇnitions of resolving power like spectral resolving power, e.g., R = =, or geometric resolving power, e.g. of microprobes, expressed by Ra = a=a (a being the area under investigation and a the resolved area element) or by the number of pixels as known from digital cameras, too. For this purpose, Kaiser [1966] proposed the measure ˘ =1=xACV being the reciprocal value of the analytical value which corresponds to the critical value (Kaiser's 7.5 Limit Values 207

\3"-limit). From today's experience, detection power, DP, should be expressed { when quantiˇed { by the reciprocal value of detection limit

DP =1=xLD (7.50) For a detection limit of 0.5 µg/g, the detection power would amount 2 g/µg(2106) and of 1 ng/g, correspondingly 1 g/ng (109) and so on4. As shown above, limits can be estimated on two ways: from blanks and from calibration data. Modern analytical methods such as spectrometry and chromatography use a third way to an increasing extent: the estimation of detection limit by means of the signal-to-noise ratio SNR. As shown in Sect. 7.1, signal-to-noise ratio S=N can be used to characterize the precision of analytical methods. Noise is a measure of the uncertainty of dynamic blank measurements (of the \background").

Starting from the standard deviation of a net signal intensity synet , see Eq. (7.3), and applying some simplifying conditions (pairwise measurement of signal y and neighboured background y0, and, therefore, ny = ny0 = n), Winefordner and Vickers [1964]; Winefordner et al. [1967] and St. John et al. [1966, 1967] derived the following expression for the critical signal-to-noise ratio:

yc y 0 (S=N)c = t1˛; = p : (7.51) sy0 2=n In St. John et al. [1966], tables of the critical SNR have been given and, in addition, the comparability of the SNR concept to that of Kaiser has been shown. From Eq. (7.51): &

yc = y 0 + t1˛; sy0 2=n =^ yBL + ksBL (7.52) can be directly derived. Similar ideas, called the SBR-RSDB concept, were developed by Boumans [1991, 1994]. In analogy to the groups of Wine- fordner and Vickers [1964], Winefordner et al. [1967], and St. John et al. [1966, 1967], detection limits, mostly on the basis of (S=N)c = 3, have been derived for a discrete measurement of background. But the main advantage of the SNR concept in modern analytical chemistry is the fact that the signal function is recorded continuously and, therefore, a large number of both background and signal values is available. As shown in Fig. 7.9, the principles of the evaluation of discrete and continuous measurement values are somewhat different. The basic measure for the estimation of the limit of detection is the conˇdence interval of the blank. It can be calculated from Eq. (7.52). For n = 10 measurements of both blank and signal values and a risk of error of ˛ =0:05 one obtains a critical signal-to- noise ratio (S=N)c = t0:95;9 =1:83 and ˛ =0:01: (S=N)c = t0:99;9 =2:82. The common value (S=N)c = 3 corresponds to a risk of error ˛ =0:05 :::0:02 in case of a small number of measurements (n =2:::5). When n 6, a 4 This may be interpreted verbally that in 2 g, 1 µg of the analyte can be detected resp. 1 µg in 1 g, or mass proportions of 2 106 resp. 109 can be detected. 208 7 Analytical Performance Characteristics

Fig. 7.9. Evaluation of blank variations in case of discrete blank yBL measurements (A) and continuous recording of the baseline (background) y0 (B)

Fig. 7.10. Critical values y c in case that the background noise is estimated from nB = 100 values and the signal value from ny = 100 (A), ny =10(B) and ny =3(C) values

signal-to-noise ratio of (S=N)c = 2 is sufˇcient to make signiˇcant differ- entiations to the background (Lieck [1998]). Modern analytical methods like spectrometry and chromatography produce a large amount of background data, sometime this is the case for signal data, too. As a rule, it can be assumed that nB = 100 :::1000 baseline values are available. Taking as a basis for computation nB = 100, in Fig. 7.10 the signal-to-noise ratios for several numbers of signal values are given (Ehrlich and Danzer [2006]). Figure 7.10 shows that in case of a large number of baseline values relative small differences between the baseline average and the critical value results. It must be considered, though, that limits derived from the SNR characterize mainly instrumental noise and do not, as a rule, include \chemical" noise, viz such variations of measurement values which come from sample inhomogeneities, sample preparations in the course of the entire analytical 7.6 Resolving Power 209 process. This should be taken into account by carrying out real repetition measurements at several samples, if it is possible, or ˇxing a higher critical value (S=N)c = 2 or 3 as be done in ofˇcial regulations, e.g. ICH [1996].

7.6 Resolving Power

In modern analytical chemistry such methods play an increasing role which have a high resolution with regard to:

Signal range and signal separation Spatial resolution of microprobes Temporal resolution of time-resolved methods Each type of resolved measurement increases the amount of information obtainable by an analytical method, namely with regard to its capability of multielemental, spatial or temporal differentiation. The analytical resolving power (signal resolving power Rz ) is determined by the size of the recording range of the signals zmax zmax and the signal resolution z. If the signal range is a linear one, as illustrated in Fig. 7.11a, the analytical resolving power is given by z z R = max min (7.53) z z In case of variable signal half-width z as shown in Fig. 7.11b the following general expression applies (Danzer [1975]):

zmax

Rz = f (z)dz (7.54)

zmin In the case of constant signal half-width it results in Eq. (7.53) whereas, in the case of constant resolution R(z), it follows that (Kaiser [1970]; Danzer [1975])

Fig. 7.11. Analytical resolving power for z = const. (a) and (z)1 = f (z)(b) 210 7 Analytical Performance Characteristics

zmax Rz = R(z)ln (7.55) zmin The analytical resolving power is applied in several analytical ˇelds in form of well-known expressions such as, e.g., spectral resolving power R = = or mass resolving power RM = M=M. Estimated values of the analytical resolving power of several analytical methods as given in Eckschlager and Danzer [1994] are shown in Table 7.6.

Table 7.6. Analytical resolving power Rz of several analytical methods

Method Rz Spot tests 1 Titration 10 Optical emission spectroscopy by grating instruments 200,000 by prism instruments 10,000 by Quantometers 60 Mass spectroscopy 500 High-resolution MS 200,000 UV-Vis spectrophotometry 50 X-ray spectrometry Wavelength-dispersive 5,000 Energy-dispersive 500 Infrared spectrometry 1,000 Gas chromatography 1,000 Capillary GC 10,000

The analytical resolving power can be interpreted as being the maximum number of signals which can ˇnd place within a given registration range. Therefore, it is evident that Rz is a measure of the multielement efˇciency of analytical methods and inuences strongly selectivity. In the same way, spatial resolving power is a measure of the efˇciency of distribution-analytical methods in micro- and surface analysis as well as scanning methods. From all the systematic representations of distribution- analytical problems given in Danzer et al. [1991], the mostly relevant are represented in Fig. 7.12. The spatial resolving power will be described for three concrete cases: (i) Lateral resolving power characterizing line scans as shown in Fig. 7.12b

l R = (7.56a) lateral l where l is the lateral resolution (e.g., spot diameter) and l the line distance under inspection. Rlateral is also relevant to point analysis 7.6 Resolving Power 211

Fig. 7.12. Relevant types of distribution analysis

(Fig. 7.12a). In the special case where l is ˇxed perpendicular to the surface of the sample the depth resolving power Rdepth = d=d is relevant which becomes important for depth proˇle and thin-ˇlm analysis. (ii) Surface resolving power (area resolving power) characterizing surface scans (Fig. 7.12c) a R = (7.56b) surface a with a being the area resolution (spot area) and a the area under inspection. Important methods for micro-, lateral- and surface analysis are microprobe techniques like EPMA and SIMS, laser vaporization and excitation techniques and analytical microscopes. (iii) Volume resolving power characterizing \volume scans" practically realized mostly by combination of lateral analytical methods and sputtering techniques or by 3D-SIMS v R = (7.56c) volume v Estimated values of the lateral resolving power of some distribution- analytical methods are shown in Table 7.7 (Eckschlager and Danzer [1994])

In analogy to the spatial resolution a temporal resolution power Rt can be deˇned: t R = (7.57) t t with t being the duration of the process under control and t the time resolution of the analytical method. The order of magnitude of the processes may be characterized by years, months and days for environmental processes, quality control of production or laboratory processes on the one hand and seconds up to femtoseconds in molecular-dynamic studies by means of ultra-short time spectroscopy. 212 References

Table 7.7. Spatial resolving power Rsurface of selected analytical methods. For reasons of comparability an area of 1 mm2 has been taken; in the case of 3-D SIMS a volume of =0:1mm3 is taken as the basis

Method Rspatial Microspark OES 1 :::100 Spark source mass spectroscopy 100 :::104 Laser OES, laser MS 100 :::104 Induced autoradiography 104 :::106 Electron probe microanalysis (EPMA) 104 :::106 4 6 Ion probe microanalysis (SIMS) Rsurface 10 :::10 8 10 3-D SIMS Rvolume 10 :::10 Electron microscopy 108 :::1011 Field ion microscopy (atom probe) 1012 :::1014

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Modern analytical chemistry has a large number of hightech methods at its disposal which are able to carry out measurements with both high reliability and high throughput. In this way, an enormous amount of data is produced daily all over the world. Apart from the fact that a large part of these data remain uninterpreted and land up in \data cemeteries", a certain part (hopefully a small part) of analytical results is inadequately evaluated and presented. This is the more regrettable as the measured results obtained from the analytical instruments are not only highly reliable but also expensive.

8.1 Presentation of Analytical Results

If it can be supposed that the measured values, y , obtained in the course of the analytical process, are both precise and accurate, then the analytical results, x, do not have automatically the same quality. For this, some preconditions must be fulˇlled, namely (1) The analytical results have to be estimated from a correct evaluation model; see Sects. 6.2.5 and 6.2.6. (2) It is desirable that the sensitivity of calibration model S = y =x is not lower than 1, otherwise the uncertainty of x will increase compared with that of y , x > y ; see Sect. 6.2.2. (3) A realistic uncertainty interval has to be estimated, namely by considering the statistical deviations as well as the non-statistical uncertainties appearing in all steps of the analytical process. All the signiˇcant deviations have to be summarized by means of the law of error propagation; see Sect. 4.2. It is in the nature of people to present results { whatever kind { with the highest possible precision. From this point of view it seems to be unnatural and absurd to collect facts which may decrease the precision and, therefore, the quality of the measured data. However, that is asked of the analyst, not only for the sake of truthfulness but also for responsible comparisons of analytical results with reference values, as will be shown in Sect. 8.2. 218 8 Presentation, Interpretation and Validation of Analytical Results Analytical results should be given always in the form

(x ˙ U (x))[x] (8.1) provided it concerns an arithmetic mean x. According to Eq. (8.1), together with the mean of the analytical value, x, its extended uncertainty, U (x), has to be reported. Naturally, the unit of the measurand, [x], must be given. The extended uncertainty U (x) is obtained by multiplying the combined uncertainty u(x) by a reasonable covering factor k; see Sect. 4.2. It is recommended, that the type of uncertainty interval should be speciˇed as being, e.g., a statistical conˇdence interval, cnf(x), prediction interval, prd(x), or extended uncertainty interval, u(x). Useful additional information may be the number n of replicate analyses from which the mean and the uncertainty has been estimated, the belonging level of signiˇcance, P =1 ˛,and{if relevant { the sort of non-statistical uncertainties that have been considered. For example, copper in sewage sludge is determined by ICP-MS. The reading (output) of the instrument may be 2.58685 for the average of duplicate analyses and 0.09513 for its conˇdence limit; see Sect. 4.1.2, Eq. (4.16), each given in µg/g. As usual, in the case of routine analyses, the conˇdence limit is calculated by means of a standard deviation that has been estimated independently from a large number of measurements. On the basis of the readings the analytical result should be given as follows: (2:59 ˙ 0:10) µg=gCu(n =2;P=0:95) or, in the case that nonstatistical uncertainties have been considered: (2:59 ˙ 0:13) µg=gCu(n =2;k=2; uncertainty considers sampling and sample preparation) In each case, the number of digits should be rounded in such a way that no insignif- icant precision is feigned. Results of ultra trace analyses are sometimes characterized by relatively high uncertainties up to more than 100%. In such cases it is not allowed that the lower uncertainty limit falls below zero. Results like, e.g.,! (0:07 ˙ 0:10) +0:10 must be replaced by such as (0:07+0:10=0:07) or 0:070:07 , respectively. That means, the total uncertainty interval (conˇdence interval, prediction interval is 0 :::0:17). In general, when the conˇdence interval includes a negative content (concentration), the result has to be given in the form +U (x) xx [x] (8.2)

Other means like the median (see Eq. (4.22)) or the geometrical mean (see Eq. (4.18)) etc. have to be reported in a similar way together with the belonging uncertainty interval, e.g.,

(medfxig˙U (medfxig))[x] (8.3) 8.2 Factual Interpretation of Analytical Results 219 Together with a median it has to be said which type of median has been computed (see the footnote in Sect. 4.1.2) and also which kind of uncertainty (derived from median absolute deviation, madfxig,orquantiles; see Danzer [1989]; Huber [1981]; Hampel et al. [1986]; Rousseeuw and Leroy [1987]). Geometrical means have unsymmetrical uncertainty intervals which are characterized by a dispersion factor (see Sect. 4.1.2, Eq. (4.20)) and a covering factor k (see Sect. 4.2). Corresponding results should be given in the form ! ! ! ˙1 xgeom kslg x xgeom U (xgeom) [x]= xgeom 10 [x]= xgeom 10 [x] (8.4a) ! That means the uncertainty interval covers the range x= U (xgeom) ::: xU (xgeom)=x=(k) :::xk. Negative values of concentration at the lower limit of uncertainty do not appear in this case. Example: Uranium has been found in wine in a concentration of 2.0 ng/L. The dispersion factor has been estimated =1:2 and the coverage factor has been chosen k = 2. Then the uncertainty factor amounts =1:45 and the analytical result has to be presented in one of the following ways: ˙1 ng { 2:0 (2:4) L U (according to Eq. (8.4)) or { (0:83 :::4:8) ng Uor L +2:80 ng { 2:01:17 L U.

8.2 Factual Interpretation of Analytical Results

Analytical investigations are always carried out to serve a deˇnite purpose. In this respect analytical results have to be evaluated and interpreted. In modern ˇelds of applications like environmental monitoring, foodstuff control, medical laboratory diagnostics etc., conclusions have to be drawn about the presence of given species and their amount as well as the exceeding of limit values or falling signiˇcantly below them. In many cases, when no extreme results (near a given limit) have been obtained, the analytical results may be reported as represented in Sect. 8.1. Attention has to be paid when results are situated nearby of critical values as it is the case with limit values of all type, especially detection limits and reference values.

8.2.1 Presentation of Results Near the Limit of Detection

The situation with detection limit has been discussed in Sect. 7.5. According to Kaiser and Specker [1956]), Kaiser [1965, 1966], Long and Wineford- ner [1983], Currie [1992, 1995, 1997], and Ehrlich and Danzer [2006] etc., the following decisions and reports are recommended: 220 8 Presentation, Interpretation and Validation of Analytical Results

(1) In the case that the measured value yi is larger than the critical measuring value yc:

yi yc :\(the analyte) A has been detected (in sample X)": As shown in Sect. 7.5, the analyte under examination is deˇnitely present in the sample (apart from a remaining risk of error ˛) when yi yc. Formally, an analytical result xi can be calculated from yi by means of the corresponding calibration function. When this result (from repeated measurements) should be reported, it must be taken into account that the relative uncertainty amounts minimally 100% (see Sect. 7.5, item (1) p. 201) and, therefore, it holds that (x ˙ x). That means, that the uncertainty interval of analytical results calculated from measured values nearby the critical value covers a range of about 0 :::2x. As additional information, the limit of quantiˇcation, xLQ, should be given. (2) No signal being signiˇcantly larger than blank signals could be found:

(3) The measured value yi exceeds the measured value at the limit of detection, yLD (MLD, see Table 7.5):

yi yc :\(the analyte) A has been detected (in sample X). The limit limit of quantiˇcation is xLQ". As in case (1), the analyte is deˇnitely detected, apart from the remaining risk of error ˛. Also in this case, an analytical result xi can be calculated formally from yi by means of the corresponding calibration function. But it must be noted that the relative uncertainty of results nearby of the detection limit amounts about 50% (see Sect. 7.5, item (2), p 201). (4) The uncertainty of the results between the detection limit and the limit of quantiˇcation decreases continuously up to the precision set in advance by the precision factor k. In reaching and exceeding the quan- tiˇcation limit, analytical results can be reported as usual; see Sect. 8.1. The problematic nature of the presentation of analytical results will be illustrated by examples of results of the Mo determination in several wine samples (ˇctional, but on the basis of own studies; see Thiel et al. [2004]; Geisler [1999]) by means of ICP-MS (nuclide Mo 95): Sample 1: Mo: not detected (detection limit DL = 47 ng/L) or Mo: < 47 ng/L Sample 2: Mo: detected (25 ˙ 25) ng/L (i.e. the concentration is slightly above of that at the critical value; CV =23:5 ng/L) Sample 3: Mo: detected (48 ˙ 24) ng/L (i.e., slightly above the detection limit) Sample 41: Mo: (3580 ˙ 370) ng/L 1 Real sample #162, see Geisler [1999] 8.2 Factual Interpretation of Analytical Results 221 8.2.2 Missing Data

There are situations in analytical chemistry where it is not advisable or not allowed to present data sets that contain zero values, \less than" values, \LTV"s, and \non detectables", \ND"s). The ˇrst one (zero data) normally come from missing data because of the breakdown of instruments or lacking samples { for whatever reason. In many cases, authorities and institutions do not accept data protocols with missing data. On the other hand, chemometrical data analyses cannot be carried out in such cases. Therefore, a lot of ideas has been suggested, how missing data can be substituted (see, e.g., Little and Rubin [1987]; Helsel [1990]; Neitzel [1996]; Allison [2002]; Howell [2002]). The mostly applied procedures are substitutions of \LTV"s and \ND"s, respectively, by concrete numerical values estimated in a meaningful way and completions of missing data by such substitutes which are the most probable data in the given situation (e.g., data lists, time series, or surface measurement nets).

Substitutions of \LTV"s and \ND" results. A meaningful value to substitute an LTV or ND is x x x = LD = CV (8.5) substit 4 2 where xLD corresponds to the 6-limit and xCV to the 3-limit; see Sect. 7.5. The reason for this substitution can be derived from Fig. 8.1 that shows an unsymmetrical distribution of analytical values, e.g., a lognormal distribution. This is the most probable distribution because the lower limit of x is zero and cannot be crossed (as it would be done in case of a normal distribution). Also the assumption of a uniform distribution, as sometimes be done, is scarcely justiˇed because the probability p(x) approximates to zero as x it does. The substitute xsubstit is characterized by a relative uncertainty of 200% (in comparison: xCV by 100%, and xLD by 50%; see Sects. 7.5 and 8.2.1, (1) and (3)).

Fig. 8.1. Most probable distribution of analytical values, x, being situated below the detection limit xLD (6-limit); xCV is the analytical value at the critical measurement value (3-limit) 222 8 Presentation, Interpretation and Validation of Analytical Results

Fig. 8.2. Substitution of values in temporal or spatial successions of analytical values: (I) random series (II) maximum expected; estimation of the substitute of x14 in enlargement: 1: extrapolated x12 and x13; 2: extrapolated x15 and x16; 3: averaged 1 and 2; x14: averaged 3 and 4 (point at the connection line between x13 and x15) (deadening)

Completion of missing data. The proceeding of completion depends on the character of the incomplete data set. In conventional data lists of random character such as a nm data list (data matrix) which contain results of several analytes (m) in diverse samples (n), missing data are frequently substituted by the mean of the respective analyte over all the samples. Another proceeding is recommended in the case of time series or lateral line scans. When the values randomly scatter and { possibly { at the same time continuously slope up or down, then the mean between the two (or three) preceding and following should be taken; see Fig. 8.2a (I). On the other hand, if an extreme value, e.g., a maximum has to be expected, then it should be interpolated after extrapolation, e.g., as illustrated in Figs. 8.2a (II) and 8.2b. Such a situation is indicated by successively increasing and then decreasing values (or vice versa), to be precise at least three each of them. In case (I), the missing values x5 may be calculated as an average of the preceded and subsequent values x4 and x6, a weighted average of four or six neighbours, or may be generated by a random number out of a given random interval. In cases of extremes (II), the missing value (here x14) may be approx- imated by means of a spline function. If x14 is generated by extrapolation, a deadening of the extrapolated value (as can be seen from Fig. 8.2b) should be applied. The extrapolation is then, e.g., carried out according to 3xm1+3xm+1xm2xm+2 xm(14) = 8 . Using more than the four neighbouring values, corresponding estimations by averaging or linear or nonlinear regression have to be carried out. 8.2 Factual Interpretation of Analytical Results 223 Completion of missing data in two-dimensional data ˇelds. The goal of investigations carrying out in form of measurement points of a network { whether it be in micro- (surface analysis) or macro dimensions (deposits) { is to detect inhomogeneities, structures, gradients, \hot spots", etc. There- fore, missing data should not be substituted by averages as in the case of random data. Total averages could simulate non-evident sub-extremes when they are situated in the surrounding of real minima or maxima. Fig- ure 3.12(b) in Sect. 3.4 may illustrate the situation. A value missed in a position where a gradient exists pointing to the maximum in the foreground at the left side should be substituted by the mean of the closely neighbouring measured points, e.g., in the data matrix ...... ::: xi1;j1 xi;j1 xi+1;j1 ::: ::: xi1;j xi+1;j ::: ::: xi1;j+1 xi;j+1 xi+1;j+1 ::: ...... this can be calculated as the average of the four nearest neighbours 1 ! x = x + x + x + x (ij)4 4 i;j 1 i 1;j i+1;j i;j+1 or of the eight neighbours 1 ! x = x + x + x + x + x + x + x + x (ij)8 8 i;j 1 i 1;j i+1;j i;j+1 i 1;j 1 i 1;j+1 i+1;j 1 i+1;j+1 or weighted, e.g., in the case of equidistant measuring points as follows: 1 ! xi;j1 + xi1;j + xi+1;j + xi;j+1 + p xi1;j1 + xi1;j+1 + xi+1;j1 + xi+1;j+1 2 x(ij) = p 8;w 4+4= 2 The decision on weighting and its type should be made according to the experimental facts. Supposed the data matrix given above is as follows then the following substitutes can be calculated: ......

::: 1:25 1:22 1:53 ::: x(ij)4 =1:265

::: 1:17 1:44 ::: x(ij)8 =1:293

::: 1:09 1:23 1:41 ::: x(ij)8;w =1:288: ...... In more complicated situations as they can be found in case of geochemical studies, sophisticated methods of spatial interpolation have been developed. In principle, global and local methods can be distinguised (Burrough [1986]). In global models (e.g., trend surface analysis, FOURIER series), each point is related to all the other points in the ˇeld of study. On the other hand, local models (e.g., spline interpolation, moving averages) estimate missing values only from neighbouring points and can, therefore, manifest 224 8 Presentation, Interpretation and Validation of Analytical Results local anomalies. Frequently used are methods like Inverse Distance Weight- ing (IDW, see Burrough and McDonnell [1998] and Kriging (Akin and Siemes [1988], Einax et al [1997]).

8.2.3 Analytical Results in Relation to Fixed Values

The relation of measured results to given values, e.g., critical levels, legally ˇxed values, regulatory limits, maximum acceptable values, is of continual relevance in analytical chemistry. In the analytical reality, the problematic nature of detection leads to the test statistics, strictly speaking to the t-test (Currie [1995, 1997]; Ehrlich and Danzer [2006]). By means of that, it is tested, if the determined analytical result is signiˇcantly different from the average blank of the critical value, respectively. In the same way, the relation of measured results to limits of various kind (critical levels, xCL) can be treated in general. The situation is shown in Fig. 8.3. Frequently, limit values are given as a numerical value without any exceeding level. Then, according to the t-statistics, a signiˇcant exceeding has to be stated if x x>xCL, as illustrated in Fig. 8.3a. On the other hand,

Fig. 8.3. Relationships between critical limits xCL and analytical results x: a the critical limit is signiˇcantly exceeded; b the result falls signiˇcantly below the limit value; c given critical exceeding limit xEL = k xCL 8.2 Factual Interpretation of Analytical Results 225 if x has to be (signiˇcantly) smaller than xCL, the condition x + xxCL indicates the exceeding of the critical level. Comparison of test values with a conventional true value (\reference value") of a (certiˇed) reference material (RM, CRM). In method development and validation of analytical procedures, the comparison of experimental results with standards of diverse kind (laboratory standards, certiˇed reference materials, primary standards) plays an essential role. The decision as to whether an experimental result hits the reference value depends not only from the result itself but also from its uncertainty interval. When working with standards, there are two cases regarding their precision: (i) The reference value is given without any uncertainty or with a very small uncertainty interval in comparison with typical experimental results of laboratories (U (xRM)U (x)exp, high quality standards like primary standards, pure elements or compounds). (ii) As a rule, the reference value is given with an uncertainty which is signiˇcant and can, therefore, not be disregarded in comparison with that of experimental results.

In case (i), the comparison of an experimental result, xexp, with the reference value, xRM, of a high quality standard is simply carried out by a speciˇed t-test according# to Eq.# (4.45): # # # # # # xexp xRM p xexp xRM t^ = nexp = (8.6) sxexp sxexp where nexp is the number of replicates which results in xexp. If the uncertainty of the reference value cannot be disregarded, the generalized t-test (Welch's TW -test; see Sect.# 4.3.4, Eq.# (4.43))# # # # # # $xexp xRM %xexp xRM T^W = = (8.7) 2 s2 2 U (xRM) xexp 2 s + + s xexp k2 nexp xRM has to be applied. The variance of the reference value can be estimated from its uncertainty interval divided by the squared coverage factor k (frequently k = 2). In case (ii), the comparison of the result experimentally found and the reference value# is carried# out by the common t-test; see Sect. 4.3.4, Eq. (4.41): # #$ xexp xRM nexp nRM t^ = (8.8) sa nexp + nRM

Normally, the averaged standard deviation sa is estimated according to Eq. (4.42). If no information is given on nRM, it can be taken from the assumption that mostly nRM 20. 226 8 Presentation, Interpretation and Validation of Analytical Results Both in cases (i) and (ii) the experimental uncertainty has to be estimated realistically as is done for the reference value, too. All the sources of variations and deviations have to be included in the calculation of the uncertainty. Efforts of analysts to shine with excellent analytical performance characteristic can have a detrimental effect as the examples in Table 8.1 demonstrate.

Table 8.1. Comparison of experimentally obtained results of copper content (µg/g) and the reference value of the material BCR277 (Institute for Reference Materials and Mea- surement, Belgium); number of repetitions nexp=10

Uncertainty x =(µg=g) x =(µg=g) Cuexp CuRM Assessment on the basis Case includes experimental certiˇed value of t-test (8.6) result ⎫ A Measurement 95:1 ˙ 3:2 ⎬ The result is false because the uncertainty interval does not B + Calibration 95:1 ˙ 5:1 ⎭ include the certiˇed value The uncertainty interval includes C + Preparation 95:1 ˙ 6:9 the certiˇed value; therefore, it cannot be considered to be false

Fig. 8.4. Found result xexp in relation to the reference value RV : a illustrates the location of RV without the uncertainty intervals A and B on the one hand and within the interval C on the other; b represents in addition the uncertainty interval of the reference value 8.2 Factual Interpretation of Analytical Results 227 In Table 8.1 three different analytical results are listed, the uncertainties of which are estimated in several ways: (A) measurement uncertainty only, as sometimes can be done in analytical practice, (B) additionally uncertainty of calibration considered, and (C) uncertainty of sample preparation included (partially nonstatistically estimated). Whereas in cases (A) and (B) the results are judged to be signiˇcantly false, in case (C) the difference is statistically not signiˇcant. The situation is illustrated in Fig. 8.4a when a comparison is carried out on the basis of the t-test (Eq. 8.6). Another assessment will be found if the uncertainty of the reference value is considered likewise and, therefore, the t-test according to Eqs. (8.7) or (8.8) is applied. The corresponding t^-values (t^A =5:96, t^B =4:00, t^C =3:02) are larger in each case than the critical t-value 2.05 (on the basis ˛ =0:05, nexp = 10, and nRM = 20). That means that the comparison will become sharper if the relatively small uncertainty interval of RV is included into the test and, therefore, the result C is assessed to be false, too.

8.2.4 Interlaboratory Studies Analytical laboratories need to check their performance with regard to the production of accurate results with satisfactory precision. The most desirable way to ensure the reliability of analytical results is the participation of laboratories into regular interlaboratory tests. An interlaboratory study has to be understood as \a study in which several laboratories measure a quantity in one or more 'dentical' portions of homogeneous, stable materials under documented conditions, the result of which are compiled into a single document" (IUPAC [1994]; Prichard et al. [2001]). There are several types of interlaboratory studies (Horwitz [1994]): (1) Method performance study: All laboratories follow the same written protocol and use the same test method to measure a quantity (usually concentration of an analyte) in sets of identical test samples. The results are used to estimate the performance characteristics of the method, which are usually within-laboratory- and between-laboratory precision and { if relevant { additional parameters such as sensitivity, limit of detection, recovery, and internal quality control parameters (IUPAC Orange Book [1997, 2000]). (2) Laboratory performance study: Laboratories use the method of their choice to measure one or more quantities on one or more homogeneous and stable test samples in order to assess the performance of the laboratory or analyst. The reported results are compared among themselves, with those of other laboratories, or with the known or assigned reference value, usually with the objective of evaluating or improving laboratory performances (IUPAC Orange Book [1997, 2000]). (3) Material certiˇcation study: Study that assigns a reference value (\true value") to a quantity (concentration or property) in the test mate- 228 8 Presentation, Interpretation and Validation of Analytical Results rial. As a rule, the participating laboratories are selected according to their competence regarding the candidate reference material (IUPAC Orange Book [1997, 2000]). Proˇciency testing is a special type of laboratory performance studies deˇned as \study of laboratory performance by means of ongoing interlaboratory test comparisons"(ISO Guide 33 [1989]; ISO/REMCO N 280 [1993]; IUPAC [1993]; Prichard et al. [2001]). Proˇciency testing is an essential part of external quality assessment schemes and performance checks. The principle of proˇciency testing schemes consists in analyzing one or more samples sent to the laboratories by an external body. The analytical results returned to the organizer are evaluated in comparison to the assigned value(s) of the sample(s). Proˇciency testing can be assessed in different ways. One of the most used evaluation system is that of so-called z-scores which are deˇned as follows (Lawn et al. [1993]; Thompson and Wood [1993]): jx x j z = lab ctv (8.9) sxctv where xlabis the mean of the laboratory, xctv the assigned value of the quantity (mostly the conventional true value) and sxctv the target standard deviation. The following convention is used:

z 2 is considered to be satisfactory. 2

8.3 Chemometrical Interpretation of Analytical Data

The ultimate purpose of analytical studies is rarely a set of analytical results (data set) but frequently chemical information about the subject under investigation. Chemical information may concern properties like: { Purity, genuineness, and activity of products { Typical pattern of particular constituents in samples characterizing their authenticity and provenance { Typical substance groups being indicators for pollution in environmental media 8.3 Chemometrical Interpretation of Analytical Data 229 { Characteristic spacial and temporal structures in objects under study, etc. Data interpretation always becomes difˇcult if several properties of various objects (samples) have to be considered simultaneously.

8.3.1 Principles of Data Analysis

Usually multivariate analytical information is represented in form of a data matrix: ⎛ ⎞ x11 x12 x1m ⎜ ⎟ ⎜ x21 x22 x2m ⎟ X =(xij)=⎝ . . . ⎠ (8.10) . . . xn1 xn2 xnm where the n rows represent the n objects under investigation which are characterized each by m features (properties, e.g. contents of various chemical constituents). Different from the principles of analytical dimensionality (see Sect. 3.4), the structure of a data set X according to Eq. (8.10) is represented in an m-dimensional space. The goals of the various methods of data analysis are: (i) Recognition of structures in multivariate data (ii) Identiˇcation, reproduction and quantitative separation of such data structures (iii) Exploration of the causes for this structures (iv) Graphical visualization of the structures in two- or three-dimensional diagrams From the viewpoint of data analysis, these objectives are achieved by means of the following fundamental methods: (i) Cluster analysis (ii) Classiˇcation methods, viz multivariate variance and discriminant analysis (MVDA) (iii) Factor analysis (Principal component analysis, PCA) (iv) Display methods In addition, methods of artiˇcial intelligence (artiˇcial neural networks and genetic algorithms) are applied. The proceeding of common methods of data analysis can be traced back to a few fundamental principles the most essential of which are dimensionality reduction, transformation of coordinates, and eigenanalysis. The principle of reduction of dimensionality will be illustrated schematically. In case that the property (+=) of an object depends mainly on one 230 8 Presentation, Interpretation and Validation of Analytical Results

Fig. 8.5. Discrimination of object quality (+=) by means of one variable x1 (a), of two variables x1 and x2 (b) which are represented as biplot in (c) from which the discriminant variable dv (discriminant function df )(d) can be derived

parameter, x1, a classiˇcation of the quality can simply be done by means of common test statistics (e.g., univariate t-test). The proceeding is shown in Fig. 8.5a. On the other hand, if two parameters, x1 and x2, inuence sig- niˇcantly the object property, then a situation as shown in Fig. 8.5b may arise. Whereas the classiˇcation of the object quality can clearly be seen in Fig. 8.5a, the classiˇcation by means of both variables, x1 and x2, seems to be chaotic. However, by using a biplot diagram (Fig. 8.5c), the situation concerning the object properties becomes clear. In addition, it can be seen that a new discriminant variable, dv = f (x1;x2), can be formed by means of which simple discriminations between the object quality (+=) can be made; see Fig. 8.5d. In this way, the discrimination problem by means of two variables which are represented in two dimensions is reduced to one dimension by means of a new variable dv = f (x1;x2). All the reductions of dimensionality, from m to graphically presentable three- or two dimensions, happen according to this principle where uncorrelated variables are generated. The basis of all data-analytical procedures is the data matrix (Eq. 8.10). In many cases the original data xij have to be transformed, either into standardized data: xij xj zij = (8.11) sj or suitable measures of multivariate distances, e.g., Euclidean distances: 8.3 Chemometrical Interpretation of Analytical Data 231 m 2 dij = (xik xjk) (8.12) k=1 The use of standardized data (variable standardization or column autoscaling, see Frank and Todeschini [1994]) results in data which are independent of the unit of measurement. Other types of standardization like object standardization, row autoscaling, or global standardization (global autoscaling, (xij x)=s) do not play a large role in data analysis. Important methods of data analysis base on evaluation of the covariance matrix (variance-covariance matrix) ⎛ ⎞ s11 s12 s1m ⎜ ⎟ ⎜ s21 s22 s2m ⎟ S =(sij)=⎝ . . . ⎠ (8.13) . . . sm1 sm2 smm which represents a symmetric matrix of pairwise covariances and variances at the diagonal; see Sect. 6.1.3. If standardized data are used the covariance matrix equals the correlation matrix ⎛ ⎞ 1 r12 r1m ⎜ ⎟ ⎜ r21 1 r2m ⎟ R =(rij)=⎝ . . . ⎠ (8.14) . . . rm1 rm2 1 The essential information on the structure of data can be extracted by means of the fundamental equation of data analysis (R I)v = 0 (8.15) where is the eigenvalue, v the eigenvector (latent vector) and I the identity matrix. Eigenvalues and eigenvectors are essential quantities for factor analysis (FA), principle component analysis (PCA) and multivariate variance and discriminant analysis (MVDA). They can be estimated by eigenanalysis (Dillon and Goldstein [1984]; Frank and Todeschini [1994]).

8.3.2 Cluster Analysis: Recognition of Inherent Data Structures

Inhomogeneities in data can be studied by cluster analysis. By means of cluster analysis both structures of objects and variables can be found without any pre-information on type and number of groupings (unsupervised learning, unsupervised pattern recognition). Geometrically illustrated, clusters are continuous regions of a high- dimensional space, each of them containing a relatively high density of points (e.g., objects), separated from each other by regions that are relatively empty (low density of points). The belonging of points (objects) to 232 8 Presentation, Interpretation and Validation of Analytical Results

Fig. 8.6. Representation of 14 multivariately characterized objects in a two-dimensional space of variables where the clusters are connected into four groups (a) and classiˇed into two differently chosen groups (b, c), respectively; d shows a nested clustering of B within A certain clusters is not unchangingly ˇxed but may alter in dependence of the method of clustering used, the measure of similarity chosen, the number of clusters supposed and factual reasons as well. An example about that is shown in Fig. 8.6. Methods of cluster analysis may be distinguished into two groups:

(i) Hierarchical clustering (ii) Non-hierarchical clusteringclustering!non-hierarchical

Hierarchical clustering can be carried out in an agglomerativeclus- tering!agglomerative and a divisiveclustering!divisive way. The results are mostly represented in form of tree-like diagrams, so-called dendrograms. The relation of the objects shown in Fig. 8.6a{c is represented in Fig. 8.7 in the form of a dendrogram. It can be seen that the objects are united at ˇrst pairwise and then step by step groupwise according to their multivariate similarity (agglomerative clustering). On the other hand, the entire population is divided stepwise into two, three, four clusters, node by node up to the single objects. 8.3 Chemometrical Interpretation of Analytical Data 233

Fig. 8.7. Schematic representation of hierarchical clustering of the 14 objects shown in Fig. 8.6; the separation lines a and b corresponds to the clusters in 8.6a,b

Fig. 8.8. Result of cluster analysis of 88 German wines according to Ward's method (Thiel et al. [2004])

There exist several methods of hierarchical clustering which use diverse measures of distance or similarity, respectively, e.g., single linkage, complete linkage, average linkage, centroid linkage, and Ward's method (Sharaf et al. [1986], Massart et al. [1988], Otto [1998]; Danzer et al. [2001]). It is possible to treat dendrograms according to graph-theoretical principles (Frank and Todeschini [1994]). However, in general, the results of clustering are evaluated qualitatively and taken as a basis of extensive studies on data structure. 234 8 Presentation, Interpretation and Validation of Analytical Results In Fig. 8.8 an example is given showing the result of cluster analysis of 88 wines on the basis of the concentrations of 13 elements (As, B, Be, Cs, Li, Mg, Pb, Si, Sn, Sr, Ti, W, and Y). The provenance of the wines was from four areas, Dienheim and In- gelheim in Rhine-Hesse and Bad Durkheim and Landau in Rhinelande- Palatinate. At ˇrst glance, two clusters can be seen; looking at them a second time, four groups of may be supposed and taken as the basis of discriminant analysis the result of which is shown in Fig. 8.11. Non-hierarchical cluster methods have in common with classiˇcation methods that pre-information on the number of classes is needed or desired to start an iteration process. In the course of the clustering a rearrangement of objects between several clusters is possible. One of the most used techniques of non-hierarchical cluster analysis is the density method (potential method). The high density of objects in the m-dimension that characterizes clusters is estimated by means of a density function (potential function) P . For this, the objects are modelled by Gaus-

Fig. 8.9. Gaussian functions with different smoothing parameters 1 and 2 (a,c) and the corresponding potential functions (b,d) 8.3 Chemometrical Interpretation of Analytical Data 235 sian functions F () with a parameter (\standard deviation", \smoothing parameter") set in advance as shown in Fig. 8.9a,c. The potential function P () results as superimposing curve of the Gaussian functions; see Fig. 8.9b,d. Whereas the superposition of the Gaussian functions Fi(1)(i = A; B; :::;G) do not produce a satisfactory separation, the Fi(2) do superimpose to three meaningful clusters. Cluster analysis is important in all situations where homogeneity of data on the one hand and latent structures on the other hand play a signiˇcant role in evaluation and interpretation of analytical results. This applies in particular for single objects with extreme properties like outliers, hot spots etc that can easily be recognized being singletons among clusters.

8.3.3 Classiˇcation: Modelling of Data Structures

Classiˇcation methods consist generally of two steps, the learning phase (training step) and the working phase (classiˇcation phase). The assignment of objects to several classes is carried out on the basis of classiˇcation rules that have to be learned by means of a set of objects { their belonging to various classes is well-known. Such classes may concern quality, sorts, origins in the widest sense (provenance, producers, counterfeits etc) where sometimes it is a matter of alternative categories like good/poor, healthy/ill, genuine/imitated etc. The basis of classiˇcation is supervised learning where a set of known objects that belong unambiguously to certain classes are analyzed. From their features (analytical data) classiˇcation rules are obtained by means of relevant properties of the data like dispersion and correlation. One of the powerful classiˇcation methods is multivariate variance and discriminant analysis (MVDA) (Dillon and Goldstein [1984]; Ahrens and Lauter [1974]; Danzer et al. [1984]). T dfj = vj xi (8.16)

By means of eigenanalysis multivariate discriminant functions, df i, can be derived with eigenvectors vj (j =1; 2;:::;p) where p m is the rank of the matrix R; see Eq. (8.14), m is the number of original variables (i =1; 2;:::;m). With Eq. (8.16) the discriminant functions are linear combinations of the original variables

dfj = 1j x1 + 2j x2 + :::+ mj xm (8.17) in which they play an especially important role { their variance within the classes is small and between the classes large. Besides, such variables are weighted slightly and are correlated with others. In this way, discriminant functions are obtained; their information content decreases in the order df1 > df2 > df3 > ::: > dfp . At best, the measures df1 and df2 can embody 236 8 Presentation, Interpretation and Validation of Analytical Results

Fig. 8.10. Schematic representation of three classes in a two-dimensional discriminant space df1 vs df2; R1, R2, R3 are the conˇdence radii of the respective classes

80{95% of the total information and, therefore, a two-dimensional plot of them may represent a realistic illustration of the respective classiˇcation as is schematically shown in Fig. 8.10. The boundaries between the classes may be estimated according to

2 2(n k)(nk +1)F1˛;f1=2;f2=nk1 Ri = (8.18) nk(n k 1) In a corresponding way, measures of other class ˇgures, e.g., ellipses, can also be calculated. Pattern recognition has been successfully applied early in the ˇelds of criminalistics (Duewer and Kowalski [1975]; Saxberg et al. [1978]), archaeology (Danzer et al. [1984, 1987]) as well as characterization of food (Forina and Armanino [1982]; Borszeki et al. [1986a]), and wine (Borszeki et al. [1986b]). Wine data of German growing areas, the clustering of which has been shown in Fig. 8.8, have been classiˇed afterwards by MVDA (Thiel et al. [2004]). The result can be seen in Fig. 8.11. Whereas the separation of wines from Bad Durkheim, Landau and In- gelheim/Dienheim succeeds satisfactorily, an overlap has to be recognised between Ingelheim and Dienheim. The total prediction rate, estimated by cross validation, is 88%. Classiˇcation of wines according to the grape variety succeeds better, in general, because there are many more typical bouquet components (several hundreds) than mineral and trace elements being typical for the origin of wine. The organic compounds can be analyzed easily and reliably by Headspace Solid-Phase Microextraction Capillary Gas Chromatography and afterwards used for classiˇcation (De la Calle et al. [1998]). An example 8.3 Chemometrical Interpretation of Analytical Data 237

Fig. 8.11. Representation of classiˇcation of 88 German wines by discriminant analysis (1st vs 2nd discriminant function) ı Bad Durkheim (Rhinelande-Palatinate) Landau (Rhinelande- Palatinate) ♦ Ingelheim (Rhine- Hesse) Dienheim (Rhine- Hesse)

Fig. 8.12. Representation of the both ˇrst discriminant functions df 1 vs df 2 obtained by discriminant analysis (MVDA) of 65 German wines from ˇve wine-growing regions according to six different grape varieties by means of 58 features (bouquet components) Muller-Thurgau ı Riesling ♦ Silvaner Scheurebe + Weissburgunder Gewurztraminer Portugieser Fig. 8.13. Discriminant analysis of 36 wines from the wine-growing region of Alzey according to four grape varieties (see above) by means of 58 features showing the classiˇcation of 65 German wines from ˇve wine-growing regions according to six grape varieties is given in Fig. 8.12 (Danzer et al. [1999]), whereas in Fig. 8.13 the discrimination of four grape varieties from only one single region (Alzey) can be seen. 238 8 Presentation, Interpretation and Validation of Analytical Results As expected, the separation improves if the sources of variation are reduced as in the situation represented in Fig. 8.13 in comparison with Fig. 8.12 where various wine-growing areas are considered. There are many classiˇcation methods apart from linear discriminant analysis (Derde et al. [1987]; Frank and Friedman [1989]; Huberty [1994]). Particularly worth mentioning are the SIMCA method (Soft independent modelling of class analogies) (Wold [1976]; Frank [1989]), AL- LOC (Coomans et al. [1981]), UNEQ (Derde and Massart [1986]), PRIMA (Juricskay and Veress [1985]; Derde and Massart [1988]), DASCO (Frank [1988]), etc. In general, the quality of classiˇcations depends on several factors such as:

{ All the essential features should be contained in the data set. { The total number of objects (n) should be large enough (as a rule n should be about 3m).

{ The number of objects in every class (nk) should be of comparable size. { The variables (features) should be normally distributed.

If normal distribution cannot be assumed, nonparametric techniques of classiˇcation should be applied. Widely used is the k-nearest neighbours method (kNN) (see Cover and Hart [1967]; Sharaf et al. [1986]; Frank and Todeschini [1994]; Danzer et al. [2001]). The kNN technique is both a very simple and powerful method of data classiˇcation. An unknown object is assigned according to the majority votes of its k nearest neighbours in the learning set in the m-dimensional space. The respective votes are given on the basis of multivariate distances, e.g., according to Eq. (8.12). In Fig. 8.14 the principle of object assignment is illustrated. To avoid ambiguities, an odd number of nearest neighbours should be chosen.

Fig. 8.14. Result of 3NN for two unknown objects A and B: A is assigned to class ı, B cannot be classiˇed signiˇcantly to any of the given classes 8.3 Chemometrical Interpretation of Analytical Data 239

Table 8.2. Euclidian distances dE and distance's ranks of an unknown wine sample to 16 samples with known growing region (GR), according to Danzer et al. [2001]

No.12345678910111213141516 GRS-US-US-URRRBBBBBR-HR-HR-HR-HR-H dE 8.73 7.85 7.74 6.11 7.13 7.27 5.18 7.37 8.18 6.01 4.86 6.55 7.63 5.95 7.26 7.39 Rank 16 14 13 57921015416123811

In the context of wine classiˇcation, an unknown wine sample had to be identiˇed as being from Saale-Unstrut (S-U), Rhineland (R), Baden (B), or Rhine-Hesse (R-H). By means of the distances and ranks given in Table 8.2, the unknown was assigned to B (by 1NN, 2NN [2B], 3NN [2B/R-H], 4NN [3B/R-H], and 5NN [3B/R-H,R]. With the development of methods of artiˇcial intelligence, neural networks are used increasingly for classiˇcation; see Sect. 8.3.6.

8.3.4 Factor Analysis: Causes of Data Structures

As a rule, the properties of objects are determined neither by only one feature nor by all the features equally. Instead the measured parameters inuence variously the properties in a complex way because the variables are usually not independent from each other but are correlated to a certain degree. The goal of factor analysis (FA) and their essential variant principal component analysis (PCA) is to describe the structure of a data set by means of new uncorrelated variables, so-called common factors or principal components. These factors characterize frequently underlying real effects which can be interpreted in a meaningful way. The principle of FA and PCA consists in an orthogonal decomposition of the original nm data matrix X into a product of two matrixes, F (nk matrix of factor scores, common factors) and L (km matrix of factor loadings) X = LF (8.19a) that means ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ x11 x1m l11 l1m f11 f1k ⎝ . . . ⎠ ⎝ . . . ⎠ ⎝ . . . ⎠ . .. . = . .. . . .. . (8.19b) xn1 xnm lk1 lkm fn1 fnk and the linear model of any xij becomes

xij = l1j fi1 + l2j fi2 + :::+ lkj fik (8.19c) This is the complete factor solution which admittedly contains uncorrelated variables but all the k factors are extracted completely and no reduction 240 8 Presentation, Interpretation and Validation of Analytical Results in dimensionality occurs. However, it is usually the aim of data analysis to reduce the variables up to the essential one on one hand and the unimpor- tant on the other. Therefore, the reduced factor solution is more important which decomposes X into L, F and an additional nm matrix, the residual matrix (error matrix) E: X = LF+ E (8.20a) ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ x11 x1m l11 l1m f11 f1q ⎝ . . . ⎠ ⎝ . . . ⎠ ⎝ . . . ⎠ . .. . = . .. . . .. . xn1 xnm lq1 lqm fn1 fnq ⎛ ⎞ (8.20b) e11 e1m ⎝ . . . ⎠ + . .. . en1 enm

xij = l1j fi1 + l2j fi2 + :::+ lqj fiq + eij (8.20c) where q = k p is the number of essential factors and p the number of insigniˇcant factors which can be transferred to the error term. By means of this reduction of dimensions the information in the form of variance is subdivided into essential contributions (common and speciˇc variance) on one hand and residual variance on the other:

var(xij )total = var(xij )comm + var(xij )spec + var(xij )res (8.21) The main difference between factor analysis and principal component analysis is the way in which the variances of Eq. (8.20) are handled. Whereas the interest of FA is directed on the common variance var(xij )comm and both the other terms are summarized as unique variance

var(xij )unique = var(xij )spec + var(xij )res and therefore

var(xij )total = var(xij )comm + var(xij )unique PCA separates a so-called true variance,

var(xij )true = var(xij )comm + var(xij )spec from the residual variance

var(xij )total = var(xij )true + var(xij )res Apart from this varied handling of variances, the proceedings of FA and PCA are similar corresponding to Eq. (8.20). Generally, factor analysis consists of two steps: (1) The factor extraction according to Eq. (8.20) in the course of which the number of common factors are estimated by rank analysis and coefˇcients of the factors (factor loadings) are calculated. 8.3 Chemometrical Interpretation of Analytical Data 241

Fig. 8.15. Plot of original data (a) and two-dimensional PC plot (b); p1 (91.4%) and p2 (5.6%) declare 97% of the total variance and, therefore, information of the data

(2) The factor rotation by which the factors are transformed into more interpretable variables and can be tested concerning hypothetical data structures, respectively. There are various techniques of factor rotation with speciˇc advantages in several ˇelds of application (Frank and Todeschini [1994]). The PCA can be interpreted geometrically by rotation of the m- dimensional coordinate system of the original variables into a new coordinate system of principal components. The new axes are stretched in such a way that the ˇrst principal component p1 is extended in direction of the maximum variance of the data, p2 orthogonal to p1 in direction of the remaining maximum variance etc. In Fig. 8.15 a schematic example is presented that shows the reduction of the three dimensions of the original data into two principal components. Figure 8.16 shows a principal component plot of that data the classiˇca- tion of which by MVDA was given in Fig. 8.11. It can be seen that a certain structure can be imagined which becomes clearer by the discrimination algorithm. An important application ˇeld of factor and principal component analysis is environmental analysis. Einax and Danzer [1989] used FA to characterize the emission sources of airborne particulates which have been sam- pled in urban screening networks in two cities and one single place. The result of factor analysis basing on the contents of 16 elements (Al, B, Ba, Cr, Cu, Fe, Mg, Mn, Mo, Ni, Pb, Si, Sn, Ti, V, Zn) determined by Optical Atomic Emission Spectrography can be seen in Fig. 8.17. In Table 8.3 the common factors, their essential loadings, and the sources derived from them are given. 242 8 Presentation, Interpretation and Validation of Analytical Results

Fig. 8.16. Representation of a principal component plot (p1 vs p2) of 88 Ger- man wines, see Fig. 8.11 and Thiel et al. [2004] ı Bad Durkheim (Rhinelande-Palatinate) Landau (Rhinelande- Palatinate) ♦ Ingelheim (Rhine- Hesse) Dienheim (Rhine- Hesse)

f3 Fig. 8.17. Representation of the factors f1 vs * f3 of two cities and one single measuring point (); according to Einax and Danzer f1 [1989]

Table 8.3. Common factors of immissions of airborne particulates during the heating period (bold: key elements); according to Einax and Danzer [1989]

Factor Variance in % Essential factor loadings by Genesis 1 34.3 Si, Mg, Pb, Ti, Mn, Fe Raised geogene material (secondary dust) 2 17.2 Al, V, Ti, Ni Industrial and urban heating systems and steam- raising plants 3 16.5 Cu, Mn, Cr, Sn, Mo, Mg Industrial immission of metallurgy and metal- processing industries 4 15.9 Zn, Ni, Ti Ubiquitous elements, sedi- mented by precipitates 8.3 Chemometrical Interpretation of Analytical Data 243 8.3.5 Exploratory Data Analysis and Display Methods: Visualization of Data Structures

Sometimes the interpretation of analytical data does not need the deepest mathematical analysis but it is sufˇcient to get an impression on the structure of the data. Although the basic idea of graphical data interpretation is ancient (e.g., Brinton [1914]), the fundamentals of modern explorative data analysis (EDA) has been developed in the 1960s (Tukey [1962, 1977]). The goal of EDA is to reveal structures, peculiarities and relationships in data. So, EDA can be seen as a kind of detective work of the data analyst. As a result, methods of data preprocessing, outlier selection and statistical data analysis can be chosen. EDA is especially suitable for interactive proceeding with computers (Buja et al. [1996]). Although graphical methods cannot substitute statistical methods, they can play an essential role in the recognition of relationships. An informative example has been shown by Anscombe [1973] (see also Danzer et al. [2001], p 99) regarding bivariate relationships. The most important methods of explorative data analysis concern the study of the distribution of the data and the recognition of outliers by boxplots (Fig. 8.18), histograms (Fig. 8.19), scatterplot matrices (Fig. 8.20), and various schematic plots.

Boxplots (Box-Whisker plots)

Fig. 8.18. Representation of some trace elements in wines in form of boxplots, constructed as follows: box: lower quartile, median , and upper quartile; whiskers: minima and maxima within box ˙1:5 of the quartiles' difference; outliers : outside of box ˙1:5of quartiles' difference (according to Danzer et al. [2001]) 244 8 Presentation, Interpretation and Validation of Analytical Results

Histograms

Fig. 8.19. Histograms of trace elements in wine for manganese (a) and copper (b) (according to Danzer et al. [2001])

Scatterplot matrixes

Fig. 8.20. Scatterplot matrix of ˇve trace elements in 19 wine samples; one of the samples is marked (+) in all the special plots to have the opportunity for comparison (according to Danzer et al. [2001])

Icon plots (schematic plots). Under this term various schematic graphs are put together (Kleiner and Hartigan [1981]; du Toit et al. [1986]; Nagel et al. [1996]) such as plots of suns, stars, glyphes, diamants, faces, owers, trees, castles, etc. From these, such graphs are privileged which easily 8.3 Chemometrical Interpretation of Analytical Data 245 can be differentiated visually. This applies particularly to faces and trees. In Fig. 8.21 some icon plots are presented to illustrate the high information and discrimination power of graphs.

Fig. 8.21. Starplots of ˇve trace elements (directed as shown below) in 19 wine samples (a) Chernoff-type faces symbolizing ˇve elements coded in the forms of the face, eyes, mouth, nose, and ears in four wine samples (b) (see Chernoff [1973]), and schematic representation of a tree plot of one sample characterized by 15 variables (c)

Andrews plots. In Andrews plots (Andrews [1972]) the features (variables set) are used as coefˇcients of a linear combination of trigonometric functions xi1 Fi(t)=p + xi2 sin(t)+xi3 cos(t)+xi4 sin(2t)+xi5 cos(2t)+ (8.22) 2 where t . Plots with similar coefˇcients results in similar curves; see Fig. 8.22.

Fig. 8.22. Andrews plot of ˇve wines, according to Danzer et al. [2001] 246 8 Presentation, Interpretation and Validation of Analytical Results

Minimal spanning trees and dendrograms. Both methods represent the transition to cluster analysis which belongs { in a wider sense { also to EDA. Dendrograms are well-known graphical representations of clustering which have been shown in Figs. 8.7 and 8.8. Minimal spanning tree (Kruskal [1956]; Prim [1957]; Cherition and Tarjan [1976]) is a method of graph- theoretical clustering by which a tree is build stepwise in such a way that, step by step, the link with the respective smallest distance is added.

8.3.6 Methods of Artiˇcial Intelligence

From the large repertoire of methods of artiˇcial intelligence (AI), viz visual perception learning, inference, problem solving, speech recognition, language understanding and translating, chess-playing, hypothesis testing and theorem proving, only a limited number of ˇelds are signiˇcant in analytical chemistry. The most important ˇelds for data interpretation are expert systems, neural networks and, partly, genetic algorithms. Expert systems (ES) are computer programs which contain heuristic knowledge basing on the experience of experts in the respective ˇeld of application. An ES consists of data bases being a collection of facts and ˇgures, on the one hand, and a system of rules of mathematical logic (e.g. in the form of Boolean algebra) on the other hand. Using principles such as combining, linking, inferring, and deciding, ES are schematically constructed as represented in Fig. 8.23. The proceeding of an ES by decision sequences may be illustrated by a (non-computerized) example from undergraduate training in analytical

Fig. 8.23. Structure of an expert system 8.3 Chemometrical Interpretation of Analytical Data 247 chemistry: the so-called \systematic separation scheme of cations" going back to Fresenius [1841]. Having a sample in aqueous solution, the follow- 2+ 2+ + ing subproceeding is applied to test if Pb , (Hg2 ), and Ag , respectively, are present in the sample.

IF HCl is added to the solution AND white precipitation appears THEN (AgCl and/or Hg2Cl2 and/or PbCl2) IF precipitation is extracted by hot H2O, afterwards solution/ˇltrate is cooled + HCl AND white precipitation (PbCl2) THEN Pb is detected ELSE no Pb2+ is contained in the sample solution IF NH4OH is added to the precipitation AND blackening appears (Hg + HgNH2Cl) THEN Hg is detected 2+ ELSE no Hg2 is contained in the sample solution IF NH4OH is added to the precipitation, then HNO3 is added to the ˇltrate AND white precipitation appears (AgCl) THEN Ag is detected ELSE no Ag+ is contained in the sample solution 2+ 2+ + ELSE neither Pb nor Hg2 nor Ag are contained in the sample solution

ES is based on comparable decision trees in computerized form. The development and arrangement is carried out by means of { Procedural computer languages (e.g., FORTRAN, PASCAL, BASIC, C/C++) by algorithmic links { Logical computer languages (e.g., PROLOG, LISP) by logical links { Shell systems in which speciˇc factual knowledge can be implemented Today, shell systems are most widely used in analytical applications because of their simplicity, exibility, and universality. The main advantage of ESs is the fact that implemented and learned knowledge easily and fault- lessly can be reproduced and copied, so that the knowledge may be available for a wide community of users or customers. An early ˇeld of application in analytical chemistry is structure elucidation. DENDRAL was one of the ˇrst ES in general, designed to the identiˇca- tion of organic compounds from mass spectrometric data (Buchanan and Feigenbaum [1978]). In the 1980s and 1990s a ood of expert systems has been developed in analytical chemistry for different types of application, viz: Structure elucidation by various molecule spectroscopic methods (X- PERT, Elyashberg et al. [1997]; MOLGEN, Benecke et al. [1997], SPECINFO, Bremser and Grzonka [1991]; Canzler and Hellen- brandt [1992]; Barth [1993]; Neudert and Penk [1996]; Schuur [1997]). 248 8 Presentation, Interpretation and Validation of Analytical Results

Interpretation of measurements of methods: X-ray uorescence spectrometry (Janssen and van Espen [1986]; Arnold et al. [1994]), X-ray diffraction spectra (Adler et al. [1993]), NMR spectra (HIPS, Wehrens et al. [1993a]), HPLC retention indices (RIPS, Wehrens [1994]), Karl Fischer titration (HELGA, Wunsch and Gansen [1989]). Evaluation of data and validation: multivariate data analysis (MULTI- VAR, Wienke et al. [1991]), evaluation of interlaboratory studies (IN- TERLAB, Wienke et al. [1991]), ruggedness expert system (RES, van Leeuwen et al. [1991]). Sampling strategies (BIAS, Wehrens et al. [1993b]). Laboratory Information Management Systems (LIMS) have been developed exclusively for practical application, viz, the organization and control of working course and information ow in the laboratory (Nilsen [1996]). LIMS comprises: { Acceptance, registration, and appointment of samples, generation of working sheets { Detailing the course of sample treatment and analytical procedures { Analyzing test samples (if necessary by several methods) { Evaluation and validation of analytical results and their uncertainty { Reporting or exporting of the results and data archiving By means of LIMS the reliability of the tests, the traceability of the results, and the security of the data can be guaranteed. In addition, the effectivity of the costs may be estimated and supply, service and maintenance of instruments managed. LIMS always has to be designed in an individual way according to its speciˇc proˇle. Information on LIMS can be found mainly on the internet (see Limsource, Lapitajs and Klinkner, etc.). Another form of artiˇcial intelligence is realized in artiˇcial neural networks (ANN). The principle of ANNs has been presented in Sect. 6.5. Apart from calibration, data analysis and interpretation is one of the most important ˇelds of application of ANNs in analytical chemistry (Tusar et al. [1991]; Zupan and Gasteiger [1993]) where two branches claim particular interest: (i) Classiˇcation of objects on the basis of multicomponent analytical data (ii) Spectra interpretation and spectrum-structure correlation Both cases can be dealt with both by supervised and unsupervised variants of networks. The architecture and the training of supervised networks for spectra interpretation is similar to that used for calibration. The input vector consists in a set of spectral features yi(zi) (e.g., intensities yi at selected wavelengths zi). The output vector contains information on the presence and absence of certain structure elements and groups ˇxed by learning rules (Fig. 8.24). Various types of ANN models may be used for spectra interpretation, viz mainly such as Adaptive Bidirectional Associa- tive Memory (BAM) and Backpropagation Networks (BPN). The correlation 8.3 Chemometrical Interpretation of Analytical Data 249

Fig. 8.24. Derivation of element composition or structural features from a measured spectrum by means of a trained ANN between spectra and structure improves by the extent of training sets. How- ever, it has to be considered that trained nets have excellent abilities in interpolation but worse in extrapolation. Applications of classical ANNs for spectra interpretation can be found in Stonham et al. [1975], Wythoff et al. [1990], Robb and Munk (1990), Allanic et al. [1992], Anker and Jurs (1992), Ball and Jurs [1993], and Boger and Karpas [1994]. In contrast to common ANNs, Kohonen networks produce self-organized topological feature maps (Kohonen [1982, 1984]). The basic idea of Koho- nen mapping is that information in data usually contains not only an algebraic but also a topological aspect. These double aspect is shown schematically in Fig. 8.25 where the data and the structure of them are composed.

Fig. 8.25. Various topological arrangement of data which are arithmetically identical 250 8 Presentation, Interpretation and Validation of Analytical Results Kohonen nets are constructed in such a way that neighbourhoods in the objects generate neighbouring points in the data projection. Kohonen nets are one-layered networks which treat the inputs in such a way that similar signals excite closely neighbouring neurons which on their part generate neighbouring points in the data projection map. A spectacular example for the efˇciency of Kohonen nets is the classiˇcation of 572 Italian oils from nine various regions according to their contents of eight different fatty acids (Forina and Armanino [1982]; Zupan and Gasteiger [1993]). Neural networks are helpful tools for chemists, with a high classiˇcation and interpretation capacity. ANNs can improve and supplement data arrangements obtained by common multivariate methods of data analysis as shown by an example of classiˇcation of wine (Li-Xian Sun et al. [1997]). Genetic algorithms (see Sect. 5.3), representing the third group of methods of artiˇcial intelligence, have their advantages in optimization, simulation and modelling (Lucasius and Kateman [1993]; Hibbert [1993]). Applications to data analysis are mostly focused on feature selection (variables reduction); see Li et al. [1992]; Fischbacher et al. [1995].

8.4 Analytical Images

Modern methods of surface microanalysis and analytical microscopy produce analytical information mainly in the form of images. Furthermore, analytical results of distribution analysis, obtained in various ways, by direct sample scanning, discontinuous sampling or even by remote sensing, may be presented by images. Analytical images represent such geometrical information that can be transferred to real sample positions given by the ˇeld of vision of the respective analytical instrument. On the other hand, with representations showing intensities or related variables in dependence of signal positions (analytical zvariables) such as wavelengths, retention values etc. (e.g., hyphenated techniques), no analytical images exist but higher-dimensional analytical information; see Sect. 3.4. The analytical information obtained as a function of sample coordinates, lx , ly and/or lz , can characterize (a) Chemical species, viz { In the form of two-dimensional images (which realize in fact three- dimensional information; see Sect. 3.4), z = f (lx ;ly ), representing so-called species images; see, e.g., Figs. 3.12c and 8.26 (O, Al, Fe, and Cu by EPMA)

{ Quasi-three-dimensional images, yz = f (lx ;ly ), representing concentration proˇles; see, e.g., Figs. 3.12a,b and 8.26 (Al, Fe and Cu by SIMS) 8.4 Analytical Images 251

Fig. 8.26. Segregation of copper in an iron copper soak alloy: MI metallographic image, AE absorbed electrons measured by EPMA, and four element-speciˇc X-ray scanning images by EPMA, below: three elemental-speciˇc relief plots by SIMS; according to Ehrlich et al. [1979]

{ Three-dimensional images, z = f (lx ;ly ;lz ), representing the presence of species in a given volume element, e.g., by SIMS, or yz = f (lx ;ly ;lz ), representing the spatial concentration distribution of a certain component in a given volume element; see, e.g., Fig. 8.27 (b) Chemical structure, mostly presented in form of two-dimensional images, S = f (lx ;ly ), as can be seen exemplary in Fig. 8.28 (c) Topological structure in form of metallographic images (see Fig. 8.26, on the left of the ˇrst row), images by electron microscopy (TEM, SEM, FEM), ion microscopy (FIM, FIM), scanning tunnelling microscopy (STM), and atomic force microscopy (AFM, AFP). By some of these methods, apart from topological structures, chemical species can also be recognized by use of special techniques, additional equipment, and advanced evaluation. For example, from metallographic (light- microscopic) images phases and, therefore, chemical compounds can 252 8 Presentation, Interpretation and Validation of Analytical Results

Fig. 8.27. Quantitative 3D analysis of Al on an integrated circuit (\pie section"); different concentrations are represented by various coded colours (with kind permission from Rudenauer and Steiger [1986])

Fig. 8.28. Tungsten crystal tip of 32 nm radius, obtained with FIM by EW Muller (Pennsylvania State University), according to Weissmantel et al. [1968]

be identiˇed by polarized radiation, etching procedures etc. On the other hand, by coupling a ˇeld ion microscope with a time-of-ight mass spectrometer with single ion sensitivity, the ˇeld ion atom probe (FIM-AP) is produced that can recognize atomic structures at solid state tips (as can be seen in Fig. 8.28) and also identify ions at certain lattice points by selective high voltage desorption. The dimensions of analytical images can cover a range of many orders of magnitude. The one extreme is given by country areas, mountain ranges, and ore deposits imaged by satellites as well as ranges in the atmosphere studied by remote sensing. The other one is characterized by the micro-, nano- and atomic structure of materials, microelectronic devices (integrated circuits) and crystals (see Figs. 8.27 and 8.28). Between these extremes, com- 8.4 Analytical Images 253 mon analytical images are situated, e.g., concerning species distribution on environmental compartments as well as materials having ordinary dimensions and being controlled by sampling and common analytical methods. Images play an important role not only in chemistry but also in medicine and biology, geology and agriculture. Image techniques can be classiˇed into three categories (Geladi et al. [1992a]: Geladi and Grahn [1996]): (i) Direct imaging using area detectors like diode arrays, CCD cameras, and video tubes (ii) Scanning by moving a pointlike detector in lx and ly direction over an object (instead of the detector alternatively the radiation source or the object may be moved) (iii) Reconstructing of latent images from data as in tomography and SIMS The most important properties of analytical images can be summarized as follows: { Images are two- or three-dimensional and, therefore, each image point (pixel or voxel) is characterized by two or three indices. { Images are limited in space, not inˇnite. { There must be some spatial autocorrelation in the image, that means that the local frequency have to be low or medium according to Fig. 2.6b,c (in contrast, in cases of zero- and high local frequencies, see Fig. 2.6a,d, there is no autocorrelation). { Spatial distances between object points, each of them characterized by given indices, can be measured and calculated. { Images have a certain resolution both in spatial distances and intensity. Image processing needs digitization where two critical aspects have to be considered. On the one hand the sampling theorem (Stanley [1975]; Rabiner and Gold [1975]), according to which at least twice of the limiting frequency (flim, highest frequency in the given image), should be used for digitization, and on the other hand the signal-to-noise ratio (see Sect. 7.1.1), to be precise the noise digitization, should be lower than the intrinsic noise (Geladi et al. [1992a]). The relationship between data, image and related techniques is shown in Fig. 8.29.

Fig. 8.29. The relationship between image and data, expressed by image processing, image analysis, data processing, and scientiˇc visualization (computer graphics), according to Geladi et al. [1992a] 254 8 Presentation, Interpretation and Validation of Analytical Results

Sample surface

Image of species A Species B Species C layer 1 layer 2 layer 3 . .

Fig. 8.30. Schematic dismantling of the m + 3 dimensions of multidimensional SIMS images; each original data layer contains ˇve dimensions A; B; C = f (lx , ly ) in the case represented here

Image acquisition as well as image analysis and image processing concern data of all ˇelds of analytical chemistry regardless of the spatial dimensions, in principle. Nevertheless, the most essential applications can be found in multidimensional micro- and nanoanalysis. This is because some of the so- called 3D techniques like SIMS (Rudenauer [1989]) and Auger microprobe (Ferguson [1989]) are characterized in fact by more than three dimensions (see Sect. 3.4). In the 3D mode SIMS, the secondary ions of various species are recorded, each of them in three spatial dimensions, as schematically represented in Fig. 8.30. As a result of 3D SIMS analysis, a stack of multivariate images is stored which can be described in case of qualitative species information by

Q = f (lx ;ly ;lz ) (8.23a) and in case of quantitative information by

yQ = f (lx ;ly ;lz ): (8.23b) Consequently, a multivariate image can be considered being a set of congruent images of the same object for various variables and, therefore, being an image the pixels (or voxels) of which are vectors instead of scalars (Geladi et al. [1992a,b]). The concrete situation as illustrated in Fig. 8.30 is characterized by A; B; C = f (lx ;ly ;lz ) and yA;yB;yC = f (lx ;ly ;lz ). From the multivariate stack of data there can be generated various one-, two- and three- dimensional (two-, three- and four-dimensional information) as, for example, shown in Fig. 8.31. Multidimensional image information can be processed in the same way as signal functions in general. In many cases, the basis of image processing is the two-dimensional Fourier analysis ! F (u;)= f (lx ;ly ) exp j2(u lx + ly ) dlx dly (8.24) an example of which is shown in Fig. 8.32. Fourier- and other transformations (e.g., Laplace-, Hadamard-, and Wavelet transformation) are the bases to transfer information complete and 8.4 Analytical Images 255

Fig. 8.31. Image processing in 3D SIMS: generation of a local depth proˇle and a transaxial layer image (a) and of a coaxial layer image,aspatial diagonal image, and a point analytical information (b); the representation is inspired from Rudenauer [1989]

Fig. 8.32. Two-dimensional Fourier transformation applied to a rectangle function shown in original 3D representation (a) and 2D contour plot (b) and as Fourier transforms (c,d), (according to Danzer et al. [2001]) reversible from an original data space into an abstract projection for the reason of data processing such as ˇltering, cross- and autocorrelation, convolution and deconvolution etc. These and other procedures of signal processing (smoothing, ampliˇca- tion) can be used analogously for common signal functions. They are used widely in digital photography to improve the quality of pictures, mainly by signal-to-noise enhancement, resolution improvement, and brightness- and contrast variation. 256 8 Presentation, Interpretation and Validation of Analytical Results

Fig. 8.33. Common PC decomposition of a data matrix Xm;n (a) and PC decomposition of a data block Xm;n;q of a multivariate image (b)

Multivariate analytical images may be processed additionally by chemometrical procedures, e.g., by exploratory data analysis, regression, classiˇca- tion, and principal component analysis (Geladi et al. [1992b]). The interpretation of a multivariate image is sometimes problematic because the cause for pictorial structures may be complex and cannot be interpreted on the basis of images of single species even if they are processed by ˇltering etc. In such cases, principal component analysis (PCA) may advantageously be applied. The principle of the PCA is like that of factor analysis which has been mathematically described in Sect. 8.3.4. It is represented schematically in Fig. 8.33. The data array Xm;n;q, which may be preprocessed by standardisation and transformation, is decomposed in a set of three-way products of score images, Tk, loading vectors, pk, and the residual, Em;n;q: X = TP+ E (8.25a)

x = Tipi + E ; (8.25b) i=1 cf. Eq. (8.20). The real dimensionality of the data is given by the rank k that separates typical structures from noise. The score matrices Ti have the same geometrical pixel arrangement as the raw images and are, therefore, called score images. Image processing has been applied in wide ˇelds of 2D- and 3D distribution analysis, e.g., in EPMA (Wolkenstein et al. [1997b]), SIMS (Rudenauer [1989]; Wolkenstein et al. [1997a]; Wickes et al. [2003]; Hutter et al. [1996]), AFM and STM (Fuchs et al. [1995]), and XPS (Artyushkova and Fulghum [2002]). References 257 References

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From the end of the 1960s up to the present, analytical chemistry has frequently been deˇned as the discipline that gains information on chemical composition and structure (Danzer et al. [1976, 1977, 1987]; Fresenius et al. [1992]); see Sect. 1.1. Consequently, analytical information has been characterized by information theory (Doerffel and Hildebrandt [1969]; Kaiser [1970]; Eckschlager [1971{1989]; Danzer [1973a,b,c, 1974, 1975a, 1975b, 1978]; Malissa et al. [1974]; Danzer and Eckschlager [1978]; Eckschlager and Danzer [1994]; Danzer and Marx [1979a,b]; Danzer et al. [1987, 1989, 1991]). Before, information theory was been applied to physical measurements (Brillouin [1963]) and in physical chemistry (Rackow [1963{1969]). The principle of chemical measurements, as realized in the analytical process (Fig. 2.1), corresponds in its signiˇcant steps to the general principle of information processing (Fig. 3.1). According to classical information theory, founded by Shannon [1948] (see also Shannon and Weaver [1949]), information is eliminated uncertainty about an occurrence or an object, obtained by a message or an experiment. Information is always bound up with signals. They are the car- riers of information in the form of deˇnite states or processes of material systems (Eckschlager and Danzer [1994]; Danzer [2004]); see Sect. 3.1.

9.1 Quantiˇcation of Information

Information, I, is the difference between information entropies, viz the a priori information entropy, H, that characterizes the uncertainty before a message is obtained or an experiment is carried out, and the a posteriori information entropy, H 0, that remains afterwards: I = H H 0 (9.1)

In the case of m discrete phenomena xi (e.g., outputs of an experiment), the information entropy is calculated by means of the respective probabilities P (xi) m m ! 1 H P (x) = P (xi)lbP (xi)= P (xi)lb (9.2) P (xi) i=1 i=1 266 9 Assessment of Analytical Information where P (xi) = 1. The binary logarithm as a measure of information has Hartley been introduced by [1928]. It holds lb a = log2 a =3:322 log10 a, the unit is then [H]=[I] = bit. On the other hand, the information entropy characterizing the uncertainty of a continuous random quantity with a probability density p(x)is given by +1 ! H(p )=H p (x) = p (x)lbp (x)dx (9.3)

1 +1 with 1 p (x)dx =1. In the case that the information of chemical results xi is imparted by signals zj , then it holds for the entropy (Eckschlager and Danzer [1994]) ! m H P (xij zj ) = P (xij zj )lbP (xij zj ) (9.4) i=1 When an unambiguous relationship exists between signal appearance and analytical result then Eq. (9.4) turns into Eq. (9.2). It is meaningful to differentiate (Meyer-Eppler [1969]; Danzer et al. [1987]; Eckschlager and Danzer [1994]) between:

(i) The speciˇc (partial) information content, Ii, of one deˇnite phenomenon or experimental result, e.g., the appearance of a given signal and, therefore, the presence of a deˇnite constituent in a sample

0 Ii = Hi + Hi (9.5)

with the speciˇc information entropies Hi = lbP (xi)

(ii) The average information content Ia of an experimental result out of a set of m possible

0 Ia = Ha + Ha (9.6)

where Ha is calculated according to Eq. (9.2) and corresponds to the weighed average of all the speciˇc information entropies Hi characterizing their mathematical expectation

(iii) The maximum (average) information content Imax

Imax = Hmax =lbm (9.7)

which is relevant if all the m results expected a priori have the same 0 probability P (xi)=1=m (in this case, the a posteriori entropy Hmax = lb1 = 0). 9.1 Quantiˇcation of Information 267

Table 9.1. Speciˇc (I+ and I) and average (Ia ) information contents (in bit) and speciˇc plausibilities (˘+ and ˘) in cases of various a priori probabilities of qualitative tests

Case Example P (x+) I+ I Ia ˘+ ˘ 1 General unknown case 0.5 1.00 1.00 1.00 1.00 1.00 2 Mn in steel 0.9 0.15 3.32 0.47 6.67 0.30 3 Silicon in secondary dust 0.99 0.014 6.64 0.078 71.43 0.15

The matter may be illustrated by the example of qualitative analysis. As a result of a speciˇc test, it is stated that a constituent searched for is present in the test sample (x+) or not (x) depending on whether a speciˇc signal is detected (z+) or not (z) as represented in Fig. 9.1. In Table 9.1 the different types of information contents are compiled, each for various a priori probabilities. If the probabilities are equal (case 1), I+, I and Ia are the same and Ia corresponds to the maximum information content, Ia = Imax. In cases 2 and 3, where the a priori probabilities are different, the expected (more probable) result yields the respective lower information contents whereas the unexpected (less probable) result manifests in high values of the speciˇc information.

In this context, the term plausibility which is sometimes used in clinical analytical chemistry, should be considered. Whereas information is formed by the unexpected, plausibility is generated even by the expected. From this point of view, plausibility has been described inversely to information, viz 1 Danzer ˘i = Ii ( [1983]), but because of the unit, plausibility should be deˇned by lb2 ˘i = (9.8) Ii Qualitative analysis may be carried out by means of visual signals (e.g., spot test reactions) or stationary signals in a given position zj of a signal function y = f (z), see Sects. 3.2, 3.3 and Fig. 9.1.

Fig. 9.1. Signal states in the case of qualitative tests (\+" means signal detected (z+) and \\ means not detected (z) 268 9 Assessment of Analytical Information

Fig. 9.2. Expectation range (a priori distribution) p0(x) and distribution of the measured values p (x) in case of normal 2 distributions N (0;0 ), N (;2)

In the case where a species has to be identiˇed from an instrumental- analytical record like Fig. 3.8 with the help of a deˇnite signal position, then the information content is determined according to Kullback's divergence measure of information (Kullback [1959]; Eckschlager and Stepanek [1985]; Eckschlager and Danzer [1994]) +1 0 p(x) I(p;p0)=H(p;p0) H (p )= p (x)lb dx (9.9) p0(x) 1 In this way, deviations can be characterized between an experimentally found distribution of measured values, p (x), and an a priori distribution p0(x), e.g., corresponding to an expected normal range of values. There are situations, especially with some spectroscopic methods, in which relations of the signal position, experimentally recorded on the one hand and theoretically expected on the other hand, may contain essential chemical information on the species (chemical shifts). In case of normally distributed a priori- and a posteriori distributions, 2 2 p (x)=N (; ) and p0(x)=N (0;0 ), Eq. (9.9) becomes 2 2 2 0 ( 0) + 0 I(p;p0)=lb + k 2 (9.10) 0 where k =1=(2 ln 2) = 0:72135. The situation is illustrated by Fig. 9.2. An example concerning the C=O stretching vibration in IR spectroscopy is given in Danzer et al. [2001], p 51. In contrast to quantitative analyses, the results of qualitative tests and of identiˇcations cannot be evaluated by means of mathematical statistics. Instead, information theory is a helpful tool to characterize qualitative analyses, in particular in case of multicomponent systems.

9.2 Information Content of Quantitative Analysis

In contrast to qualitative tests, in quantitative analysis more than the two signal states shown in Fig. 9.1 (+=) are evaluated. Depending on the un- 9.2 Information Content of Quantitative Analysis 269

Fig. 9.3. Signal levels in case of quantitative analysis certainty of the signal intensity, a certain number m of signal levels can be distinguished; see Fig. 9.3. The information content of quantitative analytical measurements is determined by three items:

(1) The expectation range of the results (a priori distribution) p0(x) (2) The uncertainty U (x) of the measured values, characterized by the a posteriori distribution p (x), where U (x)=U (y )=S; see Sect. 4.2 and Eq. (4.31) (3) The trueness of the analytical results or their inaccuracy, respectively, characterized by the bias ı = jx xtruej; see Sect. 7.1.3. Classical information theory according to Shannon [1948] and Bril- louin [1963] consider only items (1) and (2); the trueness of information has not been taken into account. In the case where no bias is relevant (x = xtrue), Eq. (9.9) becomes p (x) I(p;p0)=lb (9.11) p0(x) The a priori distribution, which reects our assumptions or preliminary information about the analyte in the sample, is frequently considered to be uniform, U (x ;x ); their probability density is given by min⎧ max ⎨⎪ 1 for xmin x xmax p (x)= xmax xmin (9.12) 0 ⎩⎪ 0 otherwise On the other hand, the a posteriori distribution, p(x), mostly can be considered to be a normal distribution, N (;2), or, in analytical reality, a 2 t-distribution, Nt (x;s ), with 270 9 Assessment of Analytical Information

Fig. 9.4. Uniform expectation range between xmin and xmax and normal- distributed (t-distributed) measured values

1 p (x)= p (9.13a) 2e or p 1 n p (x)= = (9.13b) 2x 2st1˛; respectively. The right-hand side of Eq. (9.13b) represents the reciprocal double-sided conˇdence interval that stands frequently for the uncertainty of measurement; see Sects. 4.1.2 and 4.2. With a uniform expectation range hxmin;xmaxi and the a posteriori uncertainty according to Eq. (9.13b) as shown in Fig. 9.4, the information content at Eq. (9.11) turns into p (x) x x I(p;p )=lb =lb max min 0 p (x) 2x 0 p (9.14) (x x ) n =lb max min =lbm 2st1˛;

In Eq. (9.14), m =(xmax xmin)=(2x) represents the number of signal levels that can be distinguished signiˇcantly within the expectation range (see Fig. 9.3). The same holds for the number of the levels of the chemical quantity, e.g., concentration levels. This is valid on the pre-condition that all the m signal levels are equally probable, viz m is constant over the entire measuring range and, therefore, the standard deviation s=const, too. In some cases, s and with it x are changing in dependence on the measurand x. Then the number of signiˇcantly distinguishable signal levels (and with it concentration levels) m is given by

xmax m = f (x)dx (9.15)

xmin where f (x)=x1 according to Danzer [1975a]. If s = const, it follows Eq. (9.14). On the other hand, for the frequent case of an approximately constant relative standard deviation srel = s=x = const, m becomes 9.2 Information Content of Quantitative Analysis 271

p xmax p n dx n x m = = ln max (9.16) 2srel t1˛; x 2srel t1˛; xmin xmin and the information content in case of constant relative standard deviation p n xmax I(p;p0)=lb ln : (9.17) 2srel t1˛; xmin If m = f (x) is not explicitly known, Eqs. (9.14) and (9.17) can be used as approximations depending on the size of the expectation range and pre- information on the relative constancy of s or srel. In an example it will be shown that the difference between both variants may be small. The determination of an analyte A will be considered with the following data: xmin = 1 wt%, xmax = 10%, xA =6:00%, s =0:05%, srel =0:0083, n = 10. It is obtained by: Eq. (9.14) according to s = const: p I(p;p0)=3:322 lg[(10 1) 10=(2 0:05 2:26)] = 6:98 bit

Eq. (9.17) according to srel = const: p I(p;p0)=3:322 lg[ 10=(2 0:0083 2:26) ln 10] = 7:60 bit It is customary in analytical chemistry to examine unknown samples by a screening procedure using a multicomponent method like OES or MS in case of inorganic constituents on the one hand and GC or HPLC in case of organic constituents on the other. In this way, an overview can be obtained on the type of constituents and their approximate contents. In many cases it is necessary to get a deeper insight into the sample composition. For this reason, the one or a few constituents have to determine more precisely. In such cases, the expectation range corresponds to the uncertainty range of the screening method (to the respective distribution of the values mea- 2 sured by screening) which may be, e.g., a normal distribution, N (scr;scr), 2 or Nt (xscr;sscr), respectively. The a posteriory distribution is given by the un- 2 2 certainty of the more precise determination, N (prec;prec), or Nt (xprec;sprec), respectively. In this case, Kullback's measure (Eq. 9.10) have to be applied that reads concretely: ( )2 + 2 2 scr prec scr prec scr 0 I(p;p0)=lb + k 2 (9.10 ) prec scr Examples for this proceeding are given in Danzer et al. [2001], p 55. The situation becomes more complex when aspects of the trueness of analytical results are included in the assessment. Trueness of information cannot be considered neither by the classical Shannon model nor by Kull- back's divergence measure if information. Instead, a model that takes account of three distributions, viz the uniform expectation range, p0(x), the distribution of the measured values, p (x), and that of the true value, r (x), as shown in Fig. 9.5, must be applied. 272 9 Assessment of Analytical Information

Fig. 9.5. Relationship of a uniform a priori expectation range (between xmin and xmax), normal-distributed (t-distributed) measured values, and the (conventional) true value xtrue

The occurrence of a bias ı in the measured results can be dealt with by the Kerridge{Bongard measure of inaccuracy (Kerridge [1961]; Bongard [1970]; Eckschlager and Stepanek [1985]; Eckschlager and Danzer [1994]). With the condition that is situated within the range x +3<

If the true value can be considered to be error-free (true ! 0), r (x) degenerates into a Dirac impulse N (true; 0). Considering real samples and the bias ı = true x, the estimate of Eq. (9.18) is given by p (x x ) n ı2 2 ^ max min I(r ; p;p0)=lb k 2 (9.19) 2st1˛; In the case of a large bias, the second term of Eq. (9.19) can exceed the ˇrst one and so the information content formally would become negative. Although this could be interpreted as misinformation, negative information contents are unusual in information theory itself. For this reason, Eckschlager and Stepanek [1985] introduced = I^(r ; p;p0)ifI^(r ; p;p0) 0 I = (9.20) =0 ifI^(r ; p;p0) < 0 The three-dimensional relationship between I, ı, and is represented in Fig. 9.6. Examples of the evaluation of biased analytical results have been given in Danzer et al. [2001], p 56. The consideration of both precision and trueness by means of the Kerridge-Bongard model can be generalized as follows: 9.3 Multicomponent Analysis 273

Fig. 9.6. Dependence of the information content on the bias, ı, and the standard deviation of the measured values, ,asde- scribed by the Kerridge- Bogard model (Eq. 9.19) according to Danzer et al. [2001]

(1) Positive values of information content always means a decrease of the a priori uncertainty about the measurand (2) The information content rises with the increasing width of the expectation range and with the precision of the measured values (3) A measurement procedure is able to yield information only if the produced results are sufˇciently accurate and precise, viz ı ! 0 and x < (xmax xmin) (4) Negative information content occurs if a signiˇcantly inaccurate measuring result appears. This has more serious consequences the more precise the measured result is (see Fig. 9.6) (5) The Kerridge-Bongard measure of inaccuracy expresses clearly the consequences of wrong analytical results: inaccurate results do not eliminate uncertainty but misinform the chemist and manifest the initial a priori uncertainty. The Kerridge-Bongard model of information is of great importance in quality assurance, in particular for the assessment of interlaboratory studies. Examples of the information-theoretical evaluation of analytical results within the context of interlaboratory comparisons have been given by Danzer et al. [1987, 2001], Wienke et al. [1991] and Danzer [1993].

9.3 Multicomponent Analysis

Multispecies analyses require two-dimensional analytical information y = f(x), see Sect. 3.4, mostly in the form of spectra and chromatograms. By evaluation of various signals or the entire signal function, simultaneous information on several sample components can be obtained (in the extreme case on all the constituents contained in the sample). The relevant quantity that characterizes multicomponent analyses is the information amount, 274 9 Assessment of Analytical Information M(n) being the sum of the information contents of all the n components under consideration: n M(n)= Ij (9.21) j=1 For the case that all the n constituents have similar expectation ranges, equally probable signal levels and are estimated with comparable precision, the maximum information amount becomes (Danzer et al. [1987])

M(n)max = n Imax = n lbm (9.22) In the concrete case of qualitative tests on n components, the maximum information amount is M(n)qual = n bit. As a rule, multicomponent analysis is carried out by instrumental- analytical methods that produce two-dimensional analytical information in form of signal functions. The information amount according to Eqs. (9.21) and (9.22) is based on the condition that the assignment between signals zj and events xi is unambiguous and, therefore, P (xijzj ) turns into P (xi); see Eqs. (9.4) and (9.2). Moreover, the signals selected for evaluation must be independent from each other; concerning correlated signals see Dupuis and Dijkstra [1975] and Danzer et al. [1989]. Another matter is the identiˇcation of a substance, viz the selection of one unknown component out of p which are possible. In case of equal probabilities for all the possibilities, the maximum information content results

M(1;p)max =lbp (9.23) The identiˇcation of n constituents out of p possibles in the case of p n is characterized by

M(n;p)max = n lbp (9.24) In the case where there are different probabilities for the presence of the p possibles, it holds in general that n p 1 M(n;p)= P (xj )lb (9.25) P (xj ) i=1 j=1 i If the n unknown components have to be found out from a manageable number of p possibles, instead of Eq. (9.24) the following relationship must be applied: n M(n;p)= lb(p k+1) (9.26) k=1 The information amount increases further, if the n identiˇed components additionally have to be determined quantitatively. In the simplest case, the maximum information amount according to Eq. (9.24) increases by the amount n lbm, given in Eq. (9.22) into 9.3 Multicomponent Analysis 275

M(n;p)max = n(lbp +lbm)=n lb(pm) (9.27) In most unfavourable cases of the chemical practice, not even the number n of the species is known, e.g., in so-called \general unknown" cases in toxicological analytics. Then, at ˇrst, it has to be tested by a powerful separation method, e.g. chromatography, how many components are present in the sample before they are identiˇed. For this an information part p I =lb (9.28) n n is needed and the total information amount M(n;p)total for the identiˇcation and determination of n unknown constituents (out of a set of p possibles) results: p M(n;p) = I + M(n;p)+M(n)=lb + n lb(p m) (9.29) total n n This information amount is independent of the way in which it is obtained, viz if different techniques are used to recognize the number of constituents (In), identify them and quantitatively determine them, or all of this is done by one and the same procedure like GC-MS. Examples are given in Eckschlager and Danzer [1994], p 56, and Danzer et al. [2001], p 65. From the analytical problem, it has be to distinguish between indentiˇca- tion of n unknown constituents and the analysis of n known components as characterized by Eq. (9.22). For the simultaneous analysis of n components, at least N n useful signals must be available. Instrumental-analytical methods produce as a rule two-dimensional information y = f (z), e.g., in form of spectra, chromatograms etc., as schematically shown in Fig. 7.11. The potential amount of signals, viz the number of signals that can max- imally be observed with no disturbance, is characterized by the analytical resolving power (see Sect. 7.6) Rz = N that is given by the boundaries of the registration range zmin and zmax as well as the signal half width z; see Eqs. (7.53) to (7.55) and Doerffel and Hildebrandt [1969], Kaiser [1970] and Danzer [1975a]. According to them, two neighbouring signals can be separately detected if their distance is at least z. The potential information amount Mpot of an analytical method is directly characterized by their signal resolution power (analytical resolution power) Rz = N

Mpot = N Imax = N lbm (9.30) Table 7.6 gives an overview about the analytical resolving power of various analytical techniques. Whereas some instrumental methods produce just one signal per component, N = n, e.g., chromatographic methods, some other methods such as atomic spectroscopy generate much more signals as required, N n. The surplus of signals, N n, leads to redundancy, RM ,

RM = Mpot M(n)=(N n) lbm (9.31a) 276 9 Assessment of Analytical Information frequently expressed in the form of relative redundancy, rM ,

RM n rM = =1 : (9.31b) Mpot N In more detail, the problem of redundancy is described in Danzer et al. [1989] and Eckschlager and Danzer [1994], where, in the example of qualitative evaluation in optical emission spectrography, the contrast between \helpful\ and \blank\ redundancy is illustrated. In OES, for about 80 elements there exist hundreds of thousands of spectral lines, but the spectrochemists focus their interest only on relatively few of them: the so- called main detection lines (prominent lines) according to Gerlach and Schweizer [1930], Gerlach and Gerlach [1936] and Gerlach and Riedl [1939]. These prominent lines are a set of lines associated with promoting redundancy, whereas the other lines fall into the category of blank redundancy. On the other hand, not only the enormous number of signals in multicomponent methods but also the large number of species that can be detected in highly resolved spectra and chromatograms, respectively, inuence the information amount. Therefore, Matherny and Eckschlager [1996] proposed the introduction of so-called relevancy coefˇcients, k, into the system of information-theoretical assessment. In analytical practice, the coefˇcients k can be considered as being weight factors of the information contents of the respective species with which Eq. (9.21) becomes n M(n)w = kj Ij (9.32) j=1 The values of these relevancy coefˇcients can be ˇxed depending on the practical chemical problem using static or dynamic models (Matherny and Eckschlager [1996]; Eckschlager and Stepanek [1985]; Eckschlager and Danzer [1994]).

9.4 Process and Image Analysis

In specialized studies, chemical measurements achieve particular importance by their temporal and spatial resolution. In process control, the momentary value is interesting only in such exceptional situations in which a warning or control limit is passed. Much more information is contained in the time function of an analytical system, y = f (t), or y = f (z;t), in case of multicomponent systems as shown in Fig. 3.11. The information amount of process analyses (under which term all time- dependent studies from chemical process control up to dynamic and kinetic studies are summarized) increases by the factor of time resolving power 9.4 Process and Image Analysis 277 t R = (9.33) t t compared with that of static chemical measurements. The information amount of temporally resolved measurements is given by

Mt = Rt Imax = Rt lbm (9.34) in case of single component analysis (Danzer [1978]) and

M(n)t = Rt M(n)=Rt n lbm (9.35) for multicomponent analysis. Similar is the situation in the case of geometrically resolved information (image information in the most general sense). The information amount increases in the following cases (see also Table 3.1):

Lateral proˇles, y = f (lx ), viz line scans along a line lx across the sample with y being a intensity measure or content; see, e.g., Fig. 3.11b

Concentration proˇles, y = f (lx ;ly ), on a sample surface lx ly (surface scan; see Fig. 3.12a,b)

Elemental images, z = f (lx ;ly ), with z being a component-speciˇc signal; see Fig. 3.12c

Black and white image information, b = f (lx ;ly ), where b is the number of grey levels (e.g. of photos)

Coloured image information, B = f (';lx ;ly ), where B is the number of intensity levels of ' colours Compared with that of common analyses which determine an average composition of the sample, the information amount of spatially resolved analyses increases by the factor of geometrical resolving power RA that may be in the concrete case

Lateral resolving power:

li Rlateral = (9.36a) li

where li is the entire lateral distance scanned on the sample, l the smallest resolvable lateral distance, given by the diameter of the measuring point (i is the index of the spatial direction, x, y ,orz = depth).

Surface resolving power: a R = (9.36b) surface a with a being the plane (area) of the entire sample under study, a the smallest resolvable element of plane (plane resolving limit,\pixel"; see Rudenauer [1989]). Areal resolving power plays an important role for 278 9 Assessment of Analytical Information all planar microprobe techniques like Electron Probe Micro Analysis (EPMA), Scanning Auger Microprobe (SAM) and 2D Secondary Ion Mass Spectrometry (SIMS); see Grasserbauer [1985], Grasserbauer et al. [1986a,b]; Rudenauer [1989] and Hutter et al. [1996].

Volume resolving power: R = (9.36c) volume

where is the sample volume under investigation, = lx ly lz 3 (li) smallest resolvable volume element (volume resolving limit, \voxel"; see Rudenauer [1989]). A well-known method that yields three- dimensional analytical information is the ion microprobe (SIMS).

In the special case of Eq. (9.36a) where li = lz is ˇxed perpendicularly to the surface of the sample, the depth resolving power Rdepth = lz =lz results which becomes important for depth proˇles and thin ˇlm analysis. For distribution analysis and analytical images the potential information amount increases by the factor A in case that all the distinguishable points (all the pixels or voxels, respectively) are analyzed. In the ideal case, it results in

MA = RA Imax = RA lbm (9.37) in case of single component analysis and

M(N )RA = RA M(N )=RA N lbm (9.38) for multicomponent analysis (Danzer [1974]; Danzer et al. [1991]; Eckschlager and Danzer [1994]) with a distribution-analytical method with the potential analytical resolving power N . The relationships at Eqs. (9.37) and (9.38) do not take into account a possible loss of precision with increasing geometrical resolving power as well as errors in adjusting the measuring points. Furthermore, the independence of all the RA measuring points must be presupposed for the estimation of the ideal information amount. Another limiting case is given by the real information amount of a given distribution-analytical result or problem, respectively,

M(n)a = a M(n)=a n lbm (9.39) that will be determined by the practically used geometrical resolving power, Rreal = a. Therefore, the information amount is a times greater than that of average analysis, M(n). The number a measuring points, e.g., spark spots or laser craters in Optical Emission Spectroscopy or Mass Spectrometry, are determined by a given problem, e.g., testing of homogeneity (see Danzer et al. [1991]; Eckschlager and Danzer [1994]). However, the real information amount, M(n)a, is always lower than the ideal one, M(N )RA , because a

{ em =lbm =lbm Precision efˇciency coefˇcient (m is the number of signal or concentration levels distinguishable by the method and m the number of levels needed to solve the given problem)

{ en = n=N Signal efˇciency coefˇcient (N = RZ analytical resolving power according to Eqs. (7.53) to (7.55) and Eq. (9.30), n number of components to be analysed)

{ ea = a=RA Spatial efˇciency coefˇcient (RA spatial resolving power; see Eqs. (9.36a{c), a number of spots needed to be investigated)

{ e# = #=Rt Time resolution efˇciency coefˇcient (Rt time resolving power, see Eq. (9.33), # time resolution needed)

{ et = tana=treq Time efˇciency coefˇcient (treq time required for analysis, tana real analysis time) A peripheral condition of the efˇciency coefˇcients is that they must be less than 1, otherwise the analytical problem cannot be solved. Therefore, ei is relevant for ei 1 and ei = 0 is assigned for ei > 0. Consequently, the efˇciency E according to Table 9.2 becomes 0 if at least one of the coefˇcients ei does not fulˇl the condition and, therefore the problem is insoluble (Danzer and Eckschlager [1978]; Eckschlager and Danzer [1994]). 280 References

Table 9.2. Information-theoretical performance parameters, according to Danzer and Eckschlager [1978] and Eckschlager and Danzer [1994]

Designation Quantity Unit Application in an- Note alytical process

2 Information densi- Da = Ma=a bit/cm Static detectors, a, spatial stor- 3 ty (storage density) D = M = bit/cm e.g., photographic age unit plate

Information capac- C = Mpot=t bit/s Dynamic detec- t time ity (channel capac- tors, e.g., photo- ity) multiplier

Information ow J =dMpot=dt bit/s Load of spectrom- (momentary eter channels, information Chromatography performance)

Information P = Mpot=tana bit/s Information po tana time of anal- performance bit/h wer of analytical ysis procedures k Information E = ei 1 Efˇciency test ei partial efˇ- efˇciency i=1 ciency coefˇ- cients

Information ˚ = E Mpot= bit/s/cu Performance/cost cost of an anal- proˇtability bit/h/cu relation ysis

References

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Danzer K (1978) Zur Ermittlung der Informationsmenge und des Informationsusses in der Prozeanalytik. Z Chem 18:104 Danzer K (1983) Moglichkeiten der informationstheoretischen Bewertung von Analysen- verfahren und -ergebnissen. In Muller RK (ed): Instrumentelle Analytik in der Toxikologie. Universitat Leipzig, p 26 Danzer K (1993) Information content of an analytical result and its assessment by interlaboratory studies. IDF Special Issue 9302: Analytical quality assurance and good laboratory practice in dairy laboratories. IDF, Brussels Danzer K (2004) A closer look at analytical signals. Anal Bioanal Chem 380:376 Danzer K, Eckschlager K (1978) Information efˇciency of analytical methods. Talanta 25:725 Danzer K, Marx G (1979a) Zu einigen informationstheoretischen Grundlagen der Struk- turanalytik, I. Ermittlung der Informationsmenge qualitativer Strukturaussagen. Chem Analit [Warsaw] 24:33 Danzer K, Marx G (1979b) Zu einigen informationstheoretischen Grundlagen der Struk- turanalytik, II. Die Informationsmenge quantitativer Strukturinformationen und Analy- senverfahren. Chem Analit [Warsaw] 24:43 Danzer K, Than E, Molch D (1976, 1977, 1987) Analytik. Systematischer Uberblick. Akademische Verlagsgesellschaft Geest & Portig, Leipzig, 1976; Wissenschaftliche Ver- lagsgesellschaft, Stuttgart, 1977 (2nd edn 1987) Danzer K, Eckschlager K, Wienke D (1987) Informationstheorie in der Analytik, I. Grund- lagen und Anwendungen auf Einkomponentenanalysen. Fresenius Z Anal Chem 327:312 Danzer K, Eckschlager K, Matherny M (1989) Informationstheorie in der Analytik, II. Mehrkomponentenanalyse. Fresenius Z Anal Chem 334:1 Danzer K, Schubert M, Liebich V (1991) Information theory in analytical chemistry, III. Distribution-analytical investigations. Fresenius J Anal Chem 341:511 Danzer K, Hobert H, Fischbacher C, Jagemann K-U (2001) Chemometrik. Grundlagen und Anwendungen. Springer, Berlin Heidelberg New York Doerffel K, Hildebrandt W (1969) Beurteilung von Analysenverfahren unter Einsatz der Informationstheorie. Wiss Z Techn Hochsch Leuna-Merseburg 11:30 Dupuis F, Dijkstra A (1975) Application of information theory to analytical chemistry. Identiˇcation by retrieval of gas chromatographic retention indices. Anal Chem 47:379 Eckschlager K (1971-1989) Theory of information as applied to analytical chemistry. Collect Czech Chem Commun 36: 3016; 37: 137, 1486; 38: 1330; 39: 1426, 3076; 40: 3627; 41: 1875, 2527; 42: 225, 1935; 43: 231; 44: 2373; 45: 2516; 46: 478; 47: 1195, 1580; 49: 2342; 50: 1359; 53: 1647, 3021; 54: 1770, 3031 Eckschlager K, Danzer K (1994) Information theory in analytical chemistry. Wiley, New York Eckschlager K, Stepanek V (1985) Analytical measurement and information. Letchworth, Res Studies Press Fresenius W, Malissa H Sr (eds), Grasserbauer M (coord), Cammann K, Valcarcel M, Zuckerman AM, Nan Z, Koch KH, Perez-Bustamante JA, Ortner, HM, Danzer K, Green 282 References

JD, Stulik K, Zyka J, Kuznetsov VI (1992) Analytical chemistry { today's deˇnition and interpretation. Fresenius J Anal Chem 343:809{835 Gerlach W, Gerlach W (1936) Die chemische Emissionsspektralanalyse. II. Anwendungen in Medizin, Chemie und Mineralogie. Voss, Leipzig Gerlach W, Riedl E (1939) Die chemische Emissionsspektralanalyse. III. Tabellen zur quantitativen Analyse. Voss, Leipzig Gerlach W, Schweizer E (1930) Die chemische Emissionsspektralanalyse. I. Grundlagen und Methoden. Voss, Leipzig Grasserbauer M (1985) Distribution analysis in materials research. Fresenius Z Anal Chem 322:105 Grasserbauer M, Dudek HJ, Ebel MF (1986a) Angewandte Oberachenanalyse . Springer, Berlin Heidelberg New York Grasserbauer M, Stingeder G, Potzl H, Guerrero E (1986b) Fresenius Z Anal Chem 323:421 Hartley RVL (1928) Transmission of information. Bell Syst Techn J 7:535 Hutter H, Brunner C, Nikolov S, Mittermayer C, Grasserbauer M (1996) Imaging surface spectroscopy for two- and three-dimensional characterization of materials. Fresenius J Anal Chem 355:585 Kaiser H (1970) Quantitation in elemental analysis. Anal Chem 42(2):24A, 42(4):26A Kerridge DF (1961) Inaccuaracy and inference. J R Statist Soc B23:184 Kullback S (1959) Information theory and statistics. Wiley, New York Malissa H, Rendl J, Clerc JT, Gottschalk G, Kaiser R, Schwarz-Bergkampf E, Spitzy H, Werder D, Zettler H (1974) Arbeitskreis \Automation in der Analyse", Informationstheo- rie in der Analytik. Fresenius Z Anal Chem 272:1 Matherny M, Eckschlager K (1996) Determination of relevancy and redundancy of the parameters of information theory. Chemom Intell Lab Syst 32:67 Meyer-Eppler W (1969) Grundlagen und Anwendungen der Informationstheorie. Springer, Berlin Heidelberg New York Rackow B (1963{1969) (30 papers on information-theoretical aspects of physicochemistry, e.g. introduction of the quantity \molbit", relationship between information and thermodynamical entropy, etc) (in German). Z Chem 3: 268, 316, 437, 477; 4: 36, 72, 73, 109, 155, 196, 236, 275, 311; 5: 67, 116, 159, 195, 238, 278, 434; 7: 398, 444, 472; 8: 33, 117, 157, 434; 9: 318; 11: 398, 436 Rudenauer FG (1989) Multidimensional image acquisition and processing in surface analysis. Fresenius J Anal Chem 333:308 Shannon CE (1948) A mathematical theory of communication. Bell Syst Techn J 27:379, 623 Shannon CE, Weaver W (1949) The mathematical theory of communication. Univ Illinois Press, Urbana Wienke D, Wank U, Wagner M, Danzer K (1991) MULTIVAR, PLANEX, INTERLAB { From a collection of algorithms to an expert system: statistics software written from chemists for chemists. In: Gmeling J (ed) Software development in chemistry, vol 5, p 113 Glossary of Analytical Terms

Globalization in science, technology and economy causes, amongst other things, problems in derivation, adaptation and acceptance of technical terms in general, and in Analytical Chemistry, too. Both vague and contradictory deˇnitions have been developed and established over the years. Therefore, in the last decade the efforts have been increased to harmonize the use of analytical terms. In particular international organisations like ISO, IUPAC render outstanding services to the ˇxing and harmonization of essential analytical terms. But also additional activities have been done in this ˇeld, e.g., by the Royal Society of Chemistry, the EURACHEM Education and Training Working Group and the Federation of European Chemical Societies (FECS), Division on Analytical Chemistry (DAC) as well as by publications in form of books (e.g., Prichard et al. [2001]) and other publications, e.g. the series \Glossary of Analytical Terms"(GAT) in the journal Accreditation and Quality Assurance, see GAT I to X [1996-1998], where multilingual terms are given, and Holcombe [1999/2000]. In the following, deˇnitions of essential analytical terms are compiled, if possible on the basis of international agreements. Attached are sparse references and cross-references. The symbols, being used here, means: ! \see also"(cross-references to terms, additional references as well as paragraphs, chapters, equations and ˇgures of this book), and is a warning notice.

Accuracy \Closeness of agreement between the re- ISO 3534-1 [1993] sult of a measurement and a true value ! Sect. 7.1 of the measurand". ! Trueness Do not confuse with Precision

Analysis (of a sample) Investigation of a sample to identify ! Quotation from Anal and/or determine (an) analyte(s) or as- Chem [1975] at the end of the say a material. Glossary 284 Glossary of Analytical Terms Analyte \The chemical entity being investigated According to Pritchard et al. (qualitatively or quantitatively)". [2001]

Analytical function (evaluation function) Inverse of the calibration function, x = f 1(y ), describing the dependence of the analytical values from the measured values.

Analytical method \Logical sequence of operations, de- ISO 3524-1 [1993] scribed generally, used in the perfor- ! Fig. 7.1 mance of measurements", e.g., the links of a given analytical technique with particular excitation and detection.

Analytical procedure \Set of operations, described speciˇcally, ISO 3524-1 [1993] used in the performance of particular Pritchard et al. [2001] analytical measurements according to a ! Fig. 7.1 given method".

Analytical process Logic sequence of objects linked by gen- ! Fig. 2.1 eral analytical standard operations.

Analytical result Analytical value attributed to a measur- According to ISO 3524-1 [1993] and, obtained by measurement and com- ! 8.1 pleted by information on the uncertainty of measurement.

Analytical quantity \Particular quantity subject to analytical According to ISO 3524-1 [1993] measurement". ! Measurand

Analytical technique \Generic analytical application of a sci- According to Pritchard et al. entiˇc principle". [2001] ! Fig. 7.1 Glossary of Analytical Terms 285 Assay Determination of how much of a sample ! Quotation from Anal is the material indicated by the name. Chem [1975] at the end of the Glossary ! e.g. analysis of ores Analytical value Magnitude of an analytical quantity, x, measured at test samples on the one hand and given for reference samples used for calibration on the other hand.

Background (instrumental background, background signal) Instrumental background is the null sig- ! IUPAC [1995]; Currie nal, obtained in the absence of any [1999] analyte- or interference-derived signal. ! Background may be set to zero, on the average, for certain instruments

Baseline \Summation of the instrumental back- IUPAC [1995]; Currie [1999] ground plus signals in the analyte (peak) region of interest due to interfering species".

Bias \The difference between the expectation According to GAT VIII [1997] of the test results and an accepted reference value".

Blank

A value, yB, obtained my measuring a ! IUPAC [1995]; Currie blank sample (in calibration, the inter- [1999] cept of the calibration curve is consid- ! Background ered to be equal to the blank). ! Baseline Blanks may be differentiated into instru- ! Chemical blank mental blank (background and baseline, respectively) and chemical blank (analyte blank). 286 Glossary of Analytical Terms Blank measurement Procedure by which a measured value is According to Pritchard et al. obtained with a sample in that the analyte [2001]; Taylor [1987]; Sharaf of interest is intentionally absent. et al. [1986] ! Blank sample

Blank sample A sample whose analyte concentration is According to Pritchard et al. below the limit of decision of the analyt- [2001] ical procedure being used. ! Blank measurement

Calibration Set of operations that establish, under ! ISO 3524-1 [1993] speciˇed conditions, the relationship be- ! GAT IV [1996] tween values of quantities indicated by a ! Pritchard et al. [2001] measuring system and the corresponding ! IUPAC [1998] values of quantities represented by a ma- ! Sect. 6.1 terial both in form of reference materials ! Sample domain ! Signal domain and samples. In a wider sense, calibration represents a set of operations that establish relationships between quantities in the sample domain with quantities in the signal domain, viz y = f (x) and z = f (Q).

Calibration function Equation for the estimation of the values ! Pritchard et al. [2001] of a measuring quantity from given val- ! IUPAC [1998] ues of a analytical quantity. The calibra- ! Sharaf et al. [1986] tion function may be known a priori by ! Sensitivity natural laws or estimated experimentally ! Response function by means of calibration samples. The calibration function represents that segment of the response function that is chosen for estimating the analytical value of an unknown sample. Glossary of Analytical Terms 287 Calibration samples Set of samples characterized by accurate ! Pritchard et al. [2001] and precise values of the measurand. In ! Reference material a concrete case, calibration samples may ! Certiˇed reference material be portions of (certiˇed) reference materials, in-house reference materials (laboratory standard samples), or spiked samples, and, in addition, blank samples.

Certiˇed reference material (CRM) \A reference material, accompanied by a ISO 3524-1 [1993] certiˇcate, one or more of whose prop- ! GAT IV [1996] erty values are certiˇed by a procedure ! Pritchard et al. [2001] which establishes traceability to an accurate realization of the unit in which the property values are expressed, and for which each certiˇed value is accompanied by an uncertainty at a stated level of conˇdence".

Chemical blank (analyte blank) \Blank which arises from contamination IUPAC [1995], Currie [1999] from the reagents, sampling procedure, or sample preparation steps which correspond to the very analyte being sought".

Coefˇcient of variation: The term is not recommended by IUPAC; ! Relative standard deviation

Concentration domain ! Sample domain One of the dimensions of the sample domain.

Conˇdence interval (CI) Statistical interval, e.g., of a mean, y , ! Sect. 7.5 Critical value cnf(y )=y ˙ y cnf, which express the ! uncertainty of measured values. CIs are ! Prediction interval applied for signiˇcance tests and to establish quantities for limit values (CV). 288 Glossary of Analytical Terms Conventional true value \Value attributed to a particular quantity ISO 3534-1 [1993] and accepted, sometimes by convention, ! True value as having an uncertainty appropriate for a given purpose".

Correlation Stochastic relationship between random variables in such a way that one depends on the other. The degree of relationship may be estimated by the correlation co- efˇcient.

Correlation coefˇcient

The correlation coefˇcient, rxy, is given ! Sect. 6.1.3 by the covariance of two random vari- The correlation coefˇcient ables x and y , cov(x;y)=sxy, divided by is not of any relevance in calibration, as a rule. This is be- the standard deviations sx and sy , see Eq. (6.3). The correlation coefˇcient becomes cause only the measured value r = 0 if there is no relationship between is a random variable and, in xy contrast, the analytical value is x and y , and r = ˙1 if there exist a xy a ˇxed value and not selected stringent deterministic dependence. randomly Correlation matrix Matrix formed by a set of correlation co- ! Eqs. (6.4) and (8.14) efˇcients related to m variables in multi-

variate data sets, R =(rxi;xj ). It is relevant in multicomponent analysis.

Critical value (CV) Limit in the signal domain, esti- Ehrlich and Danzer [2006]; mated from the average blank plus Currie [1999] its uncertainty, generally according ! Sect. 7.5 ! Fig. 7.8 to yc = y BL + U (y BL), in analytical chemistry frequently according to ! Decision limit yc = y BL +3sBL. If the critical value is exceeded, the respective analyte is reliably detected (except for a remaining risk of error ˛). Therefore, the CV stands for the guarantee of presence of an analyte. Glossary of Analytical Terms 289 Cross sensitivity (partial sensitivity) Dependence of the measured value (sig- ! Kaiser [1972]; Danzer nal intensity), yA, from other constituents [2001] than the analyte A, present in the mea- ! Sect. 7.2 suring sample, quantitatively expressed ! Eq. (3.11) by the respective partial differential quo- ! Sensitivity tient. ! Sensitivity matrix ! Total sensitivity

Determination Analysis of a sample to estimate quanti- ! Quotation from Anal tatively the amount (content, concentra- Chem [1975] at the end of the tion) of (an) analyte(s). Glossary

Evaluation function (analytical function) Inverse of the calibration function, x = f 1(y ), describing the dependence of the analytical values from the measured values, being so the basis of analytical evaluation.

Homogeneity A qualitative term used to describe ! Pritchard et al. [2001] that the analyte is uniformly distributed ! Inhomogeneity through the sample. The degree of homo- ! Sect. 2.1 geneity may also be characterized quan- ! Eq. (2.9) titatively as a result of a statistical test.

Hyphenated techniques Coupling of two (or more) separate an- ! Hirschfeld [1980] alytical techniques via appropriate inter- ! Kellner et al. [1998] faces and computer with the goal to ob- ! By hyphenating analytical tain faster a higher amount of informa- methods, the dimension of ana- tion on the subject under investigation. lytical information will be increased (usually by one) ! Sect. 3.4 Identiˇcation Recognizing of (an) unknown con- ! Sect. 9.3 stituent(s) in an analytical test sample. In contrast, by qualitative analysis it is tested whether (a) known constituent(s) are present or absent. 290 Glossary of Analytical Terms Imprecision A quantitative term to describe the (lack ! IUPAC [1995]; Currie of) \precision" of an analytical procedure [1999] (e.g. by standard deviation). ! Precision ! Imprecision of analytical results, see Sect. 7.1 ! Standard deviation Inaccuracy A quantitative term to describe the (lack ! IUPAC [1995]; Currie of) accuracy of an analytical procedure [1999] which comprises the imprecision and the ! Accuracy bias. Inaccuracy should not be confused with uncertainty, see IUPAC [1994a] Inhomogeneity \Term used to describe situations where ! Pritchard et al. [2001] the analyte is unevenly distributed ! Sect. 2.1 through the sample matrix". The term inhomogeneity The degree of inhomogeneity may be should not be confused with characterized quantitatively by Eq. (2.9) heterogeneity the value of which becomes negative with the transition from homogeneity to inhomogeneity.

Interlaboratory study \A study in which several laboratories IUPAC [1994b] measure a quantity in one or more iden- ! A minimum of ˇve labora- tical portions of homogeneous, stable tories should be used to provide materials under documented conditions, meaningful statistical conclu- the results of which are compiled into a sions from interlaboratory stud- single report". ies ! Sect. 8.2.4 According to the evaluation types, it is differentiated between: (1) Method-performance studies. (2) Laboratory-performance studies. (3) Material-certiˇcation studies. Glossary of Analytical Terms 291 Limit of decision (\3-limit of detection") The analytical value (e.g. the concen- Ehrlich and Danzer [2006]; tration) that corresponds to the critical IUPAC [1995]; Currie [1999] value. The limit of decision is of minor ! Sect. 7.5 importance in analytical chemistry be- ! Fig. 7.7 cause the detection at this level of con- ! Critical value centration succeeds only in 50% of all ! Detection limit The decision limit should cases. not be used as a performance characteristic of analytical methods and also not as a limit of guarantee of an analyte Limit of detection (LD)

The analytical value, xLD , that always Ehrlich and Danzer [2006]; produce a signal which can be distin- IUPAC [1995]; Currie [1999] guished from the blank (except for a re- ! Sect. 7.5 maining risk of error ˇ). ! Fig. 7.7 LD is the limit in the sample domain (an- ! Critical value alyte domain). It characterizes analytical procedures, in particular with regard to the limit concentration that can be detected. Therefore, the LD stands for the guarantee of absence of an analyte.

Limit of determination ! Limit of quantitation

Limit of quantitation (LQ)

An analytical value, xLQ, above which Ehrlich and Danzer [2006]; quantitative determinations are possi- IUPAC [1995]; Currie [1999] ble with a given minimum precision. ! Precision The condition on precision must be de- ! For factual reasons, the clared in each case. For a given preci- limit of quantiˇcation cannot be sion k = x =x , the limit of quantiˇ- lower than the limit of detection LQ LQ ! The declaration of preci- cation can be estimated by Eqs. (7.48) sion must always be given be- and (7.49). cause it is an inherent component of LQ Linear dynamic range The range of concentration in which the ! Sharaf et al. [1986] response varies linearly with the analyte concentration. 292 Glossary of Analytical Terms Linearity Ability of an analytical method to give a ! Pritchard et al. [2001] response which depends linearly on the ! Sharaf et al. [1986] analyte concentration.

Matrix All of the constituents of a sample except ! IUPAC Orange Book the analyte. The matrix is the carrier of ! [1997, 2000] the analyte. ! Analyte

Matrix effect Inuence of one or more matrix con- ! Sect. 3.5 stituent(s) on the analyte under study. ! Eqs. (3.12){(3.14); (3.16); Matrix inuences may affect the analyte (3.17) signal directly by interferences or indirectly by signal depression or ampliˇca- tion.

Measurand \Particular quantity subject to measure- ISO 3524-1 [1993] ment".

Measured result Measured value, obtained by measure- ! Sect. 8.1 ment and completed by information on ! Measured value the uncertainty of measurement. ! Uncertainty

Measured value \Outcome of an analytical measurement" ISO 3524-1 [1993] or \value attributed to a measurand". IUPAC [1995]; Currie [1999] A measured value is a \Magnitude of a ! Measuring quantity measuring quantity generally expressed as a unit of measurement multiplied by a number".

Measuring quantity \Attribute of a phenomenon . . . that may ISO 3524-1 [1993] be distinguished qualitatively and determined quantitatively". Glossary of Analytical Terms 293 Measuring sample Sample that is directly introduced into ! Sect. 2.2 analytical measurement. A measured sample is created from a test sample by conversion into a measurable form by means of a procedure of sample preparation.

Metrology \Science of measurement". ISO 3524-1 [1993]

Monitoring Continuous or repeated observation, measurement, and evaluation of a process in a certain ˇeld of application (e.g., environmental surveillance, health checking, foodstuff inspection, quality assurance in manufacturing), according to given schedules in space and time.

Multicomponent analysis (multispecies analysis) Simultaneous determination of several ! IUPAC [1995]; Currie analytes (species) by means of a multi- [1999] component sensing technique or hyphen- ! IUPAC [2004] ated techniques.

Noise Fluctuations of the baseline- or back- ! Background ground record of an (analytical) instru- ! Baseline ment. Noise do not provide meaningful ! Signal-to-noise ratio, R/N, information, on the contrary, it degrades is a measure of the quality of the quality of signals and, therefore their signals detectability. ! Sect. 7.5 ! Figs. 7.9B and 7.10 Population \Finite or inˇnite set of individuals (ob- Frank and Todeschini [1994] jects, items). A population implicitly con- ! Sample (in the statistical tains all the useful information for cal- sense) culating the true values of the population parameters", e.g., the mean and the standard deviation . 294 Glossary of Analytical Terms Precision \The closeness of agreement between in- ISO 3524-1 [1993]; GATII [1996] dependent test results obtained under ! Kaiser and Specker stipulated conditions". [1956] \The precision of a set of results of mea- In fact, standard deviation surements may be quantiˇed as a stan- characterizes imprecision dard deviation". ! Sect. 7.1.1; Eqs. (7.8) and (7.9) Prediction interval (PI) Statistical interval, e.g., of a mean, x, ! Sect. 7.5 prd(x)=x ˙ xprd, that express the un- ! Limit of detection certainty of analytical values which are ! Limit of quantiˇcation predicted on the basis of experimental ! Conˇdence interval calibration. PIs are applied for signiˇ- cance tests and to establish quantities for limit values (LD, LQ).

Proˇciency test \Study of laboratory performance by ISO Guide 33 [1989] means of ongoing interlaboratory test Pritchard et al. [2001] comparisons\. ! Interlaboratory study

Qualitative analysis Testing whether (a) known constituent(s) ! Sect. 9.3 are present or absent in test samples. In contrast, identiˇcation means recognizing of (an) unknown constituent(s) in a test sample.

Quantitative analysis Determination of the amount(s) of (an) ! Assay analyte(s) in a test sample.

Random variable A quantity that appears in a random ex- ! Sachs [1992] periment. Random variables relate events ! Frank and Todeschini into a set of values. [1994] Glossary of Analytical Terms 295 Range (in the analytical sense) \The interval between the upper and the Pritchard et al. [2001] lower concentration of the analyte in the sample for which it has been determined that the method is applicable".

Range (in the statistical sense) Difference between the greatest and the ! Pritchard et al. [2001] smallest values of a series of measure- ! Sect. 4.3.2 ments.

Recalibration Updating of a calibration model in the ! Standard operating proce- case that details of the analytical procedure dure are changed.

Reference material \A material or substance one or more ISO 3524-1 [1993] of whose property values are sufˇciently ! GAT IV [1996] homogeneous and well established to be ! Pritchard et al. [2001] used for the calibration of an apparatus, the assessment of a measurement method or for assigning values to materials".

Regression Statistical method to model a mathemat- ! Sachs [1992] ical equation that describes the relation- ! Frank and Todeschini ship between random variables (usually [1994] x and y ). The goal of regression analysis ! Regression model is both modelling and predicting. ! Sect. 6.1.3

Regression coefˇcients (regression parameter) Coefˇcients of the predictors in a regres- ! Eqs. (6.8) to (6.10) sion model, e.g., ax and bx or ay and by , respectively, in linear regression models. 296 Glossary of Analytical Terms Regression model Mathematical model that describes the ! Sachs [1992] relationship between random variables ! Frank and Todeschini (usually x and y ) by means of regres- [1994] sion coefˇcients and their uncertainties ! Sect. 6.1.3 as well as uncertainties of model and the Eq. (6.7) is not the inverse prediction. of Eq. (6.6), viz y^ ax In linear regression there are two differ- x^ = ay + by y =/ bx ent models: though Eq. (6.7) approximates that of the prediction of y from x to Eq. (6.6) in the same degree y^ = ax + bx x (6.6) as the correlation coefˇcient rxy and that of the prediction of x from y approximates to 1 x^ = ay + by y (6.7)

Relative standard deviation (RSD) Standard deviation expressed as a frac- The use of the term \co- tion of the mean srel = s=x. RSD is a efˇcient of variation"(\varia- dimensionless quantity; sometimes it is tion coefˇcient") is not recom- multiplied by 100 and expressed as a per- mended by IUPAC centage. ! IUPAC Orange Book [1997, 2000]

Reliability A qualitative term that covers precision and accuracy as well as robustness (ruggedness).

Repeatability (of results of measurements) \Closeness of the agreement between the ISO 3524-1 [1993] results of successive measurements of the ! GAT I [1996] same measurand carried out under the ! Sect. 7.1.3 same conditions of measurement" (Pre- ! Repeatability conditions cision under repeatability conditions). include the same measurement Repeatability may be expressed quanti- procedure, the same observer, the same measuring instrument, tatively in terms of suitable dispersion used under the same conditions, characteristics. the same location, and repetition over a short period of time ! Repeatability standard deviation ! Repeatability interval Glossary of Analytical Terms 297

Repeatability standard deviation (srepeat) Experimental standard deviation ob- ! Pritchard et al. [2001] tained from a series of n measurements ! The number of measure- under repeatability conditions. ments should be about n =10

Repeatability interval (repeatability limit) A conˇdence interval representing the ! Pritchard et al. [2001] maximum permitted difference between In the given formula, t1˛; is two single test results under repeatabil- the quantile of the respective ity conditions: t-distribution (the degrees of p freedom relates to the num- r = t1˛; 2 srepeat ber of replicates by which srepeat has been estimated)

Reproducibility (of results of measurements) \Closeness of the agreement between the ISO 3524-1 [1993] results of measurements of the same ! GAT I [1996] measurand carried out under changed ! Sect. 7.1.3 conditions of measurement" (Precision ! Reproducibility conditions under reproducibility conditions). are characterized by changing Reproducibility may be expressed quan- conditions such as: observer, measuring instrument, condi- titatively in terms of suitable dispersion tions of use, location, time, but characteristics. applying the same method ! Reproducibility standard deviation ! Reproducibility interval

Reproducibility standard deviation (srepro) Experimental standard deviation ob- ! Pritchard et al. [2001] tained from a series of measurements un- ! The number of measure- der reproducibility conditions. ments should be sufˇciently large to estimate a representative reproducibility standard deviation Reproducibility interval (reproducibility limit) A conˇdence interval representing the ! Pritchard et al. [2001] maximum permitted difference between ! In the given formula, t1˛; two single test results under repro- is the quantile of the t- ducibility conditions: distribution (the degrees of p freedom relates to the num- R = t1˛; 2 srepro ber of replicates by which srepro has been estimated) 298 Glossary of Analytical Terms Resolution Process by which a composite signal is ! Sect. 6.4.1 split up into individual forms. The reso- ! Sect. 7.6 lution can be related to: (i) Signal overlappings and ˇne structure (z-scale) (ii) Signals in close succession in time and space

Resolution limit The smallest difference z at which two ! Sharaf et al. [1986] adjacent signals can be separately ob- ! In case (ii) of resolution of served, i.e., their overlap does not exceed the resolution problem, t and a threshold of 50% of the individual pro- l are the crucial parameters ˇles. ! Sect. 7.6

Resolution power Ability of an analytical procedure to de- ! Sharaf et al. [1986] tect signals of small differences as sep- ! Sect. 7.6 arate signals. Resolution power is inversely proportional to resolution limit, e.g., R = z=z

Response Output of an analytical system as a reac- ! Stimulus tion to a certain stimulus. ! The output may be an observable or measurable effect Response function Relationship between the response of the ! Sharaf et al. [1986] analytical system and the amount of ana- ! Calibration function lyte. The overall response function is frequently nonlinear.

Response variable (dependent variable) ! Measuring quantity, measured value Glossary of Analytical Terms 299 Robustness Property of an analytical procedure that ! Burns et al. [2005] indicates insensitivity against changes of ! ICH [1996] known operational parameters on the re- ! Robustness may be quan- sults of the method and hence its suit- tiˇed by means of quantities ability for its deˇned purpose. characterizing signal effects ! Eq. (7.31)

Round robin test ! interlaboratory study

Ruggedness Property of an analytical procedure that ! Burns et al. [2005] indicates insensitivity against changes of ! Ruggedness may be quan- known operational variables and in ad- tiˇed by means of quantities dition any variations (not discovered in characterizing signal effects intra-laboratory experiments) which may ! Eq. (7.33) be revealed by inter-laboratory studies.

Sample (in the analytical sense) Portion of the object under study (the ! Pritchard et al. [2001] material submitted for analysis). ! There are various types of A sample consists of the analyte and the samples within given sampling matrix. schemes, e.g., bulk samples > primary samples > gross samples > subsamples > test samples > measuring samples ! Fig. 2.4 Sample (in the statistical sense) \Subset of a population that is collected Frank and Todeschini [1994] in order to estimate the properties of the underlying population", e.g., the sample parameters mean x and standard deviation s. In the ideal case of representative sampling, the sample parameter ˇt the parameter of the population and , respectively. 300 Glossary of Analytical Terms Sample domain (analyte domain) Field of analytical operation that is char- ! Fig. 2.12 acterized by samples' properties such as ! Signal domain type of analytes, Q, and their amount, xQ. The transition to signal domain is done by calibration and analytical measurement.

Sampling \Sequence of selective and non-selective ! Gy [1992] operations ending with the selection of ! Sect. 2.1 one or several test portions submitted to ! Fig. 2.4 the analytical process in their entirety. Their physical properties (maximum particle size, mass, etc) are speciˇed in the analytical procedure." Sampling covers sampling (in the narrow sense) and sample reduction.

Screening Testing of (a large number of) objects ! Sect. 1.2 in order to identify those with particu- ! Fig. 1.5 lar characteristics.

Selectivity The extent to which n given analytes can ! Kaiser [1972]; Danzer be measured simultaneously by (a least) [2001] n sensors (detecting channels) without ! Sect. 7.3 interferences by other components and, ! Eq. (7.24) therefore, can be detected and deter- Selectivity should not be mined independently and undisturbedly. merged with speciˇcity: selectivity relates to multicomponent analysis and speciˇcity to single component analysis Sensitivity \Change in the response of a measuring ISO 3435-1 [1993]; GAT VII instrument divided by the correspond- [1997] ing change in the stimulus". In analytical ! Sect. 7.2 measurements is this, in fact, the differ- ! Eq. (7.12) ential quotient of the measured value to Sensitivity should not be the analytical value. confused with limit of detection Glossary of Analytical Terms 301 Sensitivity matrix (Matrix of partial sensitivities) Matrix that contains all the sensitivities ! Kaiser [1972]; Danzer and cross sensitivities of a multicompo- [2001] nent (multidetector) analytical system. ! Sect. 7.2 ! Eq. (7.17) ! Cross sensitivity Signal \Response of a device (usually an instru- Sharaf et al. [1986] ment or a module of an instrument) to ! Sect. 3.3 certain stimuli". A signal is characterized ! Fig. 3.6 by at least three parameters: position, intensity, and width (symmetry, shape).

Signal domain (response domain) Field of analytical operation that is char- ! Fig. 2.12 acterized by signal properties such as sig- ! Sample domain nal position, z, and signal intensity, yz . The transition to sample domain is done by analytical evaluation (signal decoding).

Signal function Record of signal intensity in dependence of the signal position over a certain range of the z-scale: y = f (z).

Signal-to-noise ratio Measure of the precision of signal mea- ! Sect. 7.1, Fig. 7.2 surement, expressed mostly by the ratio ! Eqs. (7.1){(7.6) of the net signal value to a noise parameter (standard deviation or peak-to-peak distance).

Speciˇcity The extent to which one individual ana- ! Kaiser [1972]; Danzer lyte can be measured undisturbedly in a [2001] real sample by a speciˇc reagent, a par- ! Sect. 7.3 ticular sensor or a comparable speciˇc ! Eq. (7.26) measuring system. ! Selectivity Speciˇcity should not be merged with selectivity 302 Glossary of Analytical Terms Specimen Fraction of a lot (batch sample) taken Gy [1992] without respecting the rules for sampling correctness or under unknown conditions.

Standard deviation (SD) Dispersion parameter for the distribution ! IUPAC [1995]; Currie of measured values, sy , or analytical re- [1999] Sachs sults, sx , for a given sample or the pop- ! [1992] ! Dixon and Massey [1969] ulation, y and x . The SD is the square root of the variance. ! Sect. 4.1.2 ! Eqs. (4.12){(4.14) Standard error The term \standard error" is not ex- ! Sachs [1992] plicitly introduced. It is used sometimes ! Frank and Todeschini (a) synonymously for standard deviation [1994] and (b) for the residual standard devia- The term standard error tion in modelling and calibration. should be avoided

Stimulus Property of an analytical system to produce a response of an observation- or measuring system. Rousing effect of an analyte that can be characterized qualitatively and quantitatively.

Standard operating procedure (SOP) \A set of written instructions that docu- EPA [2001] ment a routine or repetitive activity fol- ! Pritchard et al. [2001] lowed by an organization". ! Fig. 7.1

Test Process of analyzing the sample to rec- ! Pritchard et al. [2001] ognize (an) analyte(s) and/or determine ! Quotation from Anal the amount(s) of (an) analyte(s). Chem [1975] at the end of the Glossary Glossary of Analytical Terms 303 Total sensitivity (total multicomponent sensitivity) Sensitivity of a multicomponent analysis. ! Sharaf et al. [1986] In the simplest case it is given by the ! Massart et al. [1988] determinant of the sensitivity matrix. ! Sect. 7.2 ! Eqs. (7.18){(7.20) Traceability \The property of a result of measurement GAT I [1996] whereby it can be related to appropri- ! ISO 3435-1 [1993] ate standards, generally international or ! All the standards used national standards, through an unbroken should have stated uncertainties chain of comparisons".

Trackability \The property of a result of a measure- GAT I [1996] ment whereby the result can be uniquely related to the sample".

True value \Value consistent with the deˇnition of ISO 3534-1 [1993] a given particular quantity" and \value GAT III [1996] which characterizes a quantity perfectly ! Conventional true value deˇned in the conditions which exist ! Sect. 7.1 when that quantity is considered".

Trueness \Closeness of agreement between the av- IUPAC Orange Book [1997, erage value obtained from a large series 2000] of test results and an accepted reference ! Codex alimentarius value". commission [1997] Trueness has been referred to as \accu- ! Sect. 7.1.3 racy of the mean". 304 Glossary of Analytical Terms Uncertainty of measurement \Parameter, associated with the result of ISO 3524-1 [1993] a measurement, that characterizes the EURACHEM [1995] dispersion of the values that could rea- GAT I [1996] sonably be attributed to the measurand". ! The uncertainty of mea- The uncertainty should combine both surement may be expressed by statistical and non-statistical contribu- the combined or extended uncertainty, u(y )orU (y ), respec- tions to the variation of the measured tively values which may occur in all steps of ! Sect. 4.2 the analytical process. ! Eqs. (4.25), (4.26) and (4.29) to (4.32) Uncertainty of an analytical result Interval, e.g., of a mean, U (x), that ex- ! Sect. 4.2 press the uncertainty of analytical values ! Eq. (4.32) considering statistical and non-statistical variations within the measurement process plus uncertainties of experimental calibration.

Validation (of an analytical method) \Process by which it is established, by USP XXII < 1225 > [1990] laboratory studies, that the performance Wegscheider [1996] characteristics of the method meet the EURACHEM [1998] requirements for the intended analytical ! Typical performance char- applications". acteristics that should be considered in the validation are: precision, accuracy, limit of detection, limit of quantitation, selectivity, range, linearity, robustness, ruggedness Variable \Characteristic of an object that may take Frank and Todeschini on any value from a speciˇed set". [1994] ! There are several types of variables, e.g., categorical, dependent and independent, experimental and theoretical, manifest and latent, random, standardized variables References 305 Variance Dispersion parameter for the distribution ! IUPAC [1995]; Currie 2 of measured values, sy , or analytical re- [1999] 2 ! Sachs [1992] sults, sx , for a given sample or the popu- Dixon Massey lation, 2 and 2. Statistically deˇned as ! and [1969] y x ! Sect. 4.1.2 the second moment about the mean. ! Eqs. (4.8); (4.10)

Working range ! Range (in the analytical sense)

Analysis, Identiˇcation, Determination, and Assay (quoted from Anal Chem [1975]) \While most chemists probably realize the difference between the terms analyze, identify, and determine, they are frequently careless when using them. Most frequently the term analysis is used when determination is meant. A study of the nomenclature problem indicates that only samples are analyzed; elements, ions, and compounds are identiˇed or determined. The difˇculty occurs when the sample is nominally an element or compound (of unknown purity). `Analysis of ...'(anelement or compound) must be understood to mean the identiˇcation or determination of im- purities. When the intent is to determine how much of such a sample is the material indicated by the name, assay is the proper word."

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Absolute measurement 35, 124 generation 46 Acceptance function 93, 94 genesis 45, 46 Accumulation 52 time characteristics 45 Accuracy 141, 183, 184, 283 Analytical technique 178, 284 Activation function 166, 168 Analytical trinity 5 Alternative hypothesis 79 Analytical value 14, 218, 285 Analysis 283, 305 Andrews plot 245 Analysis of variance 19, 21, 83, 101, 199 ANOVA: analysis of variance 19, 21, 83, one-way 102, 103 101, 199 three-way 104{107 Artiˇcial intelligence 246 two-way 21, 103{105 Artiˇcial neural networks 117, 120, 165, Analyte 23, 284 248 Analytical chemistry Assay 285, 305 deˇnition 1, 3 At-line analysis 15 development 2 Attribute testing 92 ˇeld of operation 9 Autocorrelation 21, 52 Autocorrelation function 22, 51 progress 5 repertoire 5 Background 52, 207, 285 Analytical data Background noise 208 dimensionality 59 Backpropagation networks 167, 248 Analytical evaluation 28, 36 Baseline 285 Analytical function 33, 59, 284, 289 Benninghoven matrix 30 Analytical image 59, 250, 252 Bias 66, 142, 184, 285 Analytical information 28 Blank 35, 63, 125, 202, 285 dimensionality 26 Blank correction 75 Analytical measurement 27, 28, 31 Blank measurement 286 Analytical method 284 Blank sample 286 Analytical procedure 284 Bootstrapping 149 Analytical process 13, 14, 284 Boxplots 243 Analytical quantity 32, 124, 284 Box-Wilson optimization method 115 Analytical resolving power 209, 275 Bulk analysis 6 Analytical result 36, 217, 218, 284 Analytical sciences 4 Calibration 28, 67, 123{125, 286 Analytical signal 2, 4, 44, 101 by artiˇcial neural networks 165 appearance 46 individual three-dimensional 147 conversion 47 linear 130 detection 47 orthogonal 140 310 Index

Calibration function 29, 34, 35, 124, 126, Covariance matrix 128, 129, 231 127, 186, 286 Coverage factor 78, 218 Calibration model 125 Critical level 224 general three-dimensional 126 Critical limit 224 linear 130 Critical sample mass 19, 20 Calibration points 125 Critical signal-to-noise ratio 207 Calibration samples 125, 287 Critical value 90, 201, 202, 204, 209, 220, Categorical quantities 28 288 Cause-effect diagrams 76 Cross sensitivity 61, 188, 193, 289 Certiˇed reference material 68, 125, 141, Cross validation 163 148, 287 Cross-correlation function 50 Chemical blank 287 Cusum control 97 Chemical information 43 Chemical measurement process 13, 43, Data bases 36 44 Data evaluation 67 Chemical shift 60 Data interpretation 229 Chemometrics 4 Data matrix 229 Chernoff face 245 Data processing 253 Classiˇcation 229 Decoding 29, 44, 123 Classiˇcation methods 235 Deˇnitive measurement 35, 124 Cluster analysis 229, 231, 234, 246 Dendrogram 232, 246 Density function 234 Clustering Depth resolving power 211, 278 hierarchical 232 Detection capability 206 Coding 28, 29, 44, 123 Detection limit 63, 207 Combined signal 48 Detection power 206 Combined uncertainty 36, 76, 218 Determination 289, 305 Comparison Deviation of standard deviations 82 additive 67 of measured values 82 multiplicative 67 Composite sample 16 Digital ˇltering 52 Concentration function 16{18 Dimensionality 53, 56, 58 Concentration proˇle 250, 277 Dimensionality reduction 229 Concordancy rate 87 Dimensions Condition number 159, 162 chemical 53 Conˇdence band 136 measuring 53 Conˇdence interval 66, 72, 73, 127, 134, physical 53 136, 147, 218, 287 statistical 53 Conˇdence limit 73, 93 Direct calibration 158 Control limit 96 Direct imaging 253 Control parameters 98 Discriminant function 230, 235 Conventional true value 183, 225, 288 Discriminant variable 230 Convolution 30 Dispersion factor 74 Correlation 22, 288 Distribution analysis 6 Correlation analysis 127 Dynamic analysis 6 Correlation coefˇcient 50, 77, 128, 129, 288 Eigenanalysis 229 Correlation matrix 128, 129, 231, 288 Eigenvalue 159, 231 Correlation time 22 Eigenvector 231 Covariance 75, 77 Elastic interactions 27 Index 311

Element analysis 29 Hidden layer 165 Elemental images 277 Histograms 243, 244 Encoding 28 Homogeneity 16{18, 20, 104, 289 Enrichment 24 Homogeneity value 20 Error of the ˇrst kind 87, 91 Homoscedasticity 129, 131, 143 Error of the second kind 79, 87, 91 Hyphenated techniques 25, 289 Error propagation 67, 75 Hypothesis testing 62 Euclidean distance 230 Evaluation 123 Icon plot 244 Evaluation function 34, 49, 126, 127, 284, Identiˇcation 7, 34, 124, 274, 289, 305 289 Image analysis 253, 276 Exceeding limit 224 Image information 277 Experimental calibration 124 Image processing 253, 255, 256 Experimental design 62, 108, 162 Imprecision 183, 290 Expert system 36, 246 In-line analysis 14, 15 Extended uncertainty 77, 203, 218 Inaccuracy 184, 269, 290 Extraction 24 Indirect calibration 150, 157, 158 Inelastic interactions 27 Factor analysis 229, 231, 239 Inuence factor 60, 61, 101 Factor extraction 240 Information 43 Factor loadings 239 one-dimensional 55 Factor rotation 241 Information amount 26, 47, 274, 278 Factor scores 239 potential 275 Factorial design Information capacity 280 complete 109, 114 Information content 266, 267, 270, 271, fractional 111 273 partial 111 of qualitative analysis 267, 268 two-level 109 Information density 280 False negative rate 87 Information efˇciency 279, 280 False positive rate 87 Information ow 279, 280 False-positive 79 Information performance 279, 280 Fourier transformation 50 Information process 43, 44 two-dimensional 254, 255 Information proˇtability 279, 280 Frequency domain 50 Inhomogeneity 16, 17, 20, 290 F-test 18{20, 81, 102, 104 Inhomogeneity variance 18 Fuzzy set theory 37, 150 Input layer 165 Inspection analysis 54 Genetic algorithms 117, 118, 120, 246, Interferences 62, 101, 154, 189 250 Interlaboratory comparisons 62 Geometrical mean 73, 74, 218, 219 Interlaboratory study 179, 195, 227, 228, Geometrical resolving power 277 273, 290 Global optimization 116, 118 Interquantile ranges 74 Global optimum 113, 116, 118 Intrinsic energy 27, 30 Gradient analysis 21 Inverse calibration 157, 159 Grid experiments 114 k nearest neighbours 238 Gross sample 16 Kinetic energy 28 Kohonen network 249 Hat matrix 163 Heterogeneity 16, 17 Laboratory information management Heteroscedasticity 132, 137 systems 248 312 Index

Laboratory performance study 227 Measuring sample 14, 23, 24, 26, 27, 147, Laboratory sample 15 293 Laboratory standards 125 Median 74, 218 Latent information 28, 43 Median absolute deviation 74 Latent variable 160 Median statistics 145 Lateral proˇles 277 Method performance study 227 Lateral resolving power 210, 211, 277 Metrology 4, 293 Latin square design 108 Micro analysis 6 Learning rate 167 Minimal spanning tree 246 Least squares Missing data linear 137 completion 222, 223 ordinary 131{134 Monitoring 21, 293 orthogonal 132 Moving averages 223 weighted 132 Multi-range calibration 151 Least trimmed squares 145 Multi-stage techniques 25 Level of signiˇcance 218 Multicollinearities 156, 160 Leverage points 144 Multicomponent analysis 6, 155, 189, Library search 36 273, 293 Limit of decision 291 Multicomponent calibration 155, 157 Limit of detection 6, 201, 204, 205, 220, Multicomponent sensitivity 188, 189 291 Multifactorial design 62, 108 multivariate 164 Multilayer perceptron 166 Limit of determination 206, 291 Multisignal calibration 152, 154 Limit of discrimination 88 Multisignal evaluation 187 Limit of quantiˇcation 201, 206, 220 Multispecies analysis 293 Limit of quantitation 291 Multivariate analytical images 256 Limit of speciˇcation 88, 89 Multivariate calibration 157, 162 Limit values 201 uncertainty 159 LIMS 248 validation 162 Line analysis 58 Multivariate discriminant function 235 Linear calibration 130, 134 Multivariate distances 230 Linear discriminant analysis 238 Multivariate image 254 Linear dynamic range 291 Multivariate similarity 232 Linearity 142, 292 Multivariate variance and discriminant Local analysis 6 analysis 229, 231, 235 Local frequency 18 Local optimum 116 Nano analysis 6 Mass resolving power 210 Natural computation 116 Material certiˇcation study 227 Natural design 116 Matrix 23, 60, 61, 292 Net function 166 Matrix effect 112, 292 Neural networks 246 Matrix inuence 148 Neuron 165, 167 Matrix of partial sensitivities 188 Noise 52, 293 Mean (mean value) 69, 71, 98 Noise function 52 Measurand 124, 292 Nonlinear calibration 134 Measured result 292 Nonlinear calibration functions 152 Measured value 14, 292 Normal distribution 69, 70 Measuring function 29, 47 logarithmic 73 Measuring quantity 292 Null hypothesis 79, 102 Index 313

Off-line analysis 14 Qualitative analysis 7, 34, 58, 124, 294 On-line analysis 14, 15 Quality assurance 90, 91, 273 Optimization 111, 112 Quality control chart 96 Optimum 114 Quality criteria 90 Order of instruments 59 Quantitative analysis 58, 124, 294 Output function 167 Output layer 165 Radial basis function nets 168 Random deviations 65{67 Partial least squares 152, 160 Random samples 69 Partial sensitivity 61, 289 Random variable 294 Random variations 69 Pattern recognition 21, 236 Range 98, 295 Peak-to-peak noise 181 Recalibration 295 Performance characteristic curve 89 Recovery 68 Performance characteristics 177 Recovery function 68, 126, 127, 141, 164 Performance parameters 5 Recovery rate 98 Pixel 277 Recovery study 141 Plausibility 267 Reduction of dimensionality 230 Poisson distribution 74 Redundancy 275 Population 293 Reference calibration 133 Potential function 234 Reference data 36 Power spectrum 51 Reference material 225, 295 Precision 63, 65, 141, 142, 162, 184, 294 Reference measurements 35 of a complete analytical technique direct 124 180 indirect 124 of a measuring system 180 Reference method 133 of an analytical procedure 179, 184 Reference spectra 36, 49 of an analytical result 180, 184 Reference value 225, 226 of analytical methods 179 Regression 127, 295 of analytical techniques 179 Regression analysis 21, 127, 138 of calibration 142 Regression coefˇcient 129, 295 of trace analyses 182 Regression model 130, 296 Precision analysis 6 Rejection function 93, 94 Prediction band 136 Relative standard deviation 72, 296 Prediction interval 127, 137, 180, 218, Reliability 296 294 Repeatability 65, 69, 179, 296 Prediction rate 88 Repeatability standard deviation 297 Predictive residual sum of squares 163 Reproducibility 65, 179, 297 Prevalency 87 Reproducibility interval 297 Primary sample 16 Reproducibility standard deviation 297 Principal component analysis 140, 152, Resampling techniques 204 160, 187, 229, 231, 239 Residual error 104 Principal component plot 241 Residual standard deviation 135 Principal component regression 160 Resolution 298 Probability 86 Resolution limit 298 conditional 86 Resolution power 209, 298 Process analysis 6, 58, 276 Response domain 301 Process sampling 21 Response function 298 Proˇciency test 228, 294 Response surface 113, 114, 117 Propagation function 166 Response variable 298 314 Index

Reverse calibration 157 two-dimensional 54 Robust calibration 133, 134, 144, 146 Signal generation 14, 43, 60 Robust regression 144 Signal half width 47, 51, 209 Robustness 63, 112, 195, 196, 198, 299 Signal intensity 28, 50, 51, 60, 61 Root mean standard error of prediction Signal model 62 171 Signal parameters 51, 60 Round robin test 299 Signal position 28, 36, 50, 51, 60 Ruggedness 63, 195, 197, 198, 299 Signal recordings 49 Signal resolving power 47, 209 Sample 299 Signal validation 44 Sample domain 28, 29, 32, 33, 201, 300 Signal-to-noise ratio 52, 180{182, 184, Sample preparation 23, 24, 67 207, 208, 301 chemical 24 Signiˇcance level 203 Sample pretreatment Simplex optimization 115, 117 physical 24 Simulated annealing 117 Sampling 15, 67, 300 Slope 125 representativeness 19 Soft modeling methods 160 Sampling theorem 253 Spatial interpolation 223 Scanning 253 Spatial resolving power 210, 212 Scattering factor 74 Species images 250 Scatterplot matrix 243, 244 Speciˇcity 63, 189{193, 195, 301 Schematic plot 243, 244 Speciˇcity function 195 Score images 256 Specimen 302 Screening 9, 85, 271, 300 Spectra interpretation 248 Screening limit 88 Spectral resolving power 210 Screening test 85, 86, 89 Spline function 151 Selectivity 159, 162, 189{193, 195, 300 Spline interpolation 223 Semivariogram analysis 23 Sensitivity 35, 61, 63, 125, 185, 186, 191, Standard 34 199, 217, 300 Standard addition method 133, 146, 147 Sensitivity coefˇcient 35 Standard additions 68 Sensitivity factor 124 Standard deviation 69, 72, 74, 75, 98, 302 Sensitivity functions 152 of noise 181 Sensitivity matrix 158, 188, 192, 301 Standard error 302 Separation 24 Standard error of prediction 162 Sequential analysis 93, 94 Standard normal distribution 70 Signal 43, 301 Standard operating procedure 180, 302 ˇne structure 52, 60 Standard samples 141 form 48 Standard value 90 hidden 46 Standardized data 230 latent 45 Standards 125 manifest 45 Stimulus 302 pattern 48 Stochastic process 22, 23 shape 52 Stochastic sampling 21 Signal averaging 52 Structure analysis 7, 9, 29, 124 Signal background 51 Subsample 16 Signal domain 28, 29, 32, 33, 47, 201, 301 Supervised learning 235 Signal function 30, 33, 44, 45, 47, 48, Surface analysis 58 50{52, 58, 180, 207, 301 Surface resolving power 211, 277 three-dimensional 55 Systematic deviations 66, 67 Index 315

Target value 90, 95, 98 Uncertainty 17, 18, 36, 65, 67, 76, 204, t-distribution 70 205, 269 Temporal resolution power 211 at critical value 204 Test 302 at limit of detection 205 for normal distribution 80 at limit of quantiˇcation 205 for outliers 81 of an analytical result 304 for trend 80 Uncertainty interval 66, 78, 202, 217, 225, statistical 78 227 Test efˇciency 87 unsymmetrical 219 Test kits 86 Uncertainty of measurement 75, 180, Test portion 16 304 Test sample 15, 16, 24, 26 Uncertainty of sampling 19 Test spectra 49 Unsupervised pattern recognition 231 Theory of fractals 59 Thresholds 201 Validation 68, 90, 140, 190, 304 Time domain 50 Validation function 141 Time resolving power 276 Variable 304 Time series analysis 21 Variable testing 92 Tolerance limits 92 Variance 71, 102, 104, 305 Total sensitivity 303 common 240 Trace analysis 6 residual 240 Traceability 125, 126, 303 true 240 Trackability 303 unique 240 Trans-line analysis 15 Variance function 144 Transfer function 166, 167 Variance-covariance matrix 231 Trend surface analysis 223 Volume resolving power 211, 278 True negative rate 86 Voxel 278 True positive rate 86 True signal function 30, 52 Warning limit 96 True value 66, 183, 303 Wavelength selection 120 Trueness 164, 183, 184, 269, 303 Weighted calibration 144 t-test Working range 305 generalized 83 Ultra trace analysis 6 z-scores 228